2205.13147

# Matryoshka Representation Learning > Equal contribution – AK led the project with extensive support from GB and AR for experimentation. ## Abstract Learned representations are a central component in modern ML systems, serving a multitude of downstream tasks. When training such representations, it is often the case that computational and statistical constraints for each downstream task are unknown. In this context, rigid fixed-capacity representations can be either over or under-accommodating to the task at hand. This leads us to ask: can we design a flexible representation that can adapt to multiple downstream tasks with varying computational resources? Our main contribution is <details> <summary>x1.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) which encodes information at different granularities and allows a single embedding to adapt to the computational constraints of downstream tasks. ${\rm MRL}$ minimally modifies existing representation learning pipelines and imposes no additional cost during inference and deployment. ${\rm MRL}$ learns coarse-to-fine representations that are at least as accurate and rich as independently trained low-dimensional representations. The flexibility within the learned ${\rm Matryoshka~Representations}$ offer: (a) up to $\mathbf{14}\times$ smaller embedding size for ImageNet-1K classification at the same level of accuracy; (b) up to $\mathbf{14}\times$ real-world speed-ups for large-scale retrieval on ImageNet-1K and 4K; and (c) up to $\mathbf{2}$ % accuracy improvements for long-tail few-shot classification, all while being as robust as the original representations. Finally, we show that ${\rm MRL}$ extends seamlessly to web-scale datasets (ImageNet, JFT) across various modalities – vision (ViT, ResNet), vision + language (ALIGN) and language (BERT). ${\rm MRL}$ code and pretrained models are open-sourced at https://github.com/RAIVNLab/MRL. ## 1 Introduction Learned representations [57] are fundamental building blocks of real-world ML systems [66, 91]. Trained once and frozen, $d$ -dimensional representations encode rich information and can be used to perform multiple downstream tasks [4]. The deployment of deep representations has two steps: (1) an expensive yet constant-cost forward pass to compute the representation [29] and (2) utilization of the representation for downstream applications [50, 89]. Compute costs for the latter part of the pipeline scale with the embedding dimensionality as well as the data size ( $N$ ) and label space ( $L$ ). At web-scale [15, 85] this utilization cost overshadows the feature computation cost. The rigidity in these representations forces the use of high-dimensional embedding vectors across multiple tasks despite the varying resource and accuracy constraints that require flexibility. Human perception of the natural world has a naturally coarse-to-fine granularity [28, 32]. However, perhaps due to the inductive bias of gradient-based training [84], deep learning models tend to diffuse “information” across the entire representation vector. The desired elasticity is usually enabled in the existing flat and fixed representations either through training multiple low-dimensional models [29], jointly optimizing sub-networks of varying capacity [9, 100] or post-hoc compression [38, 60]. Each of these techniques struggle to meet the requirements for adaptive large-scale deployment either due to training/maintenance overhead, numerous expensive forward passes through all of the data, storage and memory cost for multiple copies of encoded data, expensive on-the-fly feature selection or a significant drop in accuracy. By encoding coarse-to-fine-grained representations, which are as accurate as the independently trained counterparts, we learn with minimal overhead a representation that can be deployed adaptively at no additional cost during inference. We introduce <details> <summary>x2.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) to induce flexibility in the learned representation. ${\rm MRL}$ learns representations of varying capacities within the same high-dimensional vector through explicit optimization of $O(\log(d))$ lower-dimensional vectors in a nested fashion, hence the name ${\rm Matryoshka}$ . ${\rm MRL}$ can be adapted to any existing representation pipeline and is easily extended to many standard tasks in computer vision and natural language processing. Figure 1 illustrates the core idea of ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) and the adaptive deployment settings of the learned ${\rm Matryoshka~Representations}$ . <details> <summary>x3.png Details</summary>  ### Visual Description ## Diagram: Adaptive Retrieval and Classification System with Hierarchical Training ### Overview The diagram illustrates a two-phase system: **Inference** (top) and **Training** (right). The Inference phase includes components for shortlisting, re-ranking, adaptive retrieval, and adaptive classification. The Training phase depicts a hierarchical loss structure with decreasing granularity (e.g., `L(z1:d/16)`, `L(z1:d/8)`, etc.) and a final aggregated loss `L(z)`. A circular element with a cross and a stylized figure (red base, yellow top) connects the phases. --- ### Components/Axes 1. **Inference Phase**: - **Shortlisting**: Initial candidate selection (orange arrow). - **Re-ranking**: Refines shortlisted candidates (gray arrow). - **Adaptive Retrieval**: Processes re-ranked candidates (blue arrow). - **Adaptive Classification**: Outputs hierarchical confidence scores (bar chart with red, blue, yellow segments). 2. **Training Phase**: - **Hierarchical Loss Stack**: Vertical column with labeled loss functions: - `L(z1:d/16)` (red) - `L(z1:d/8)` (orange) - `L(z1:d/4)` (blue) - `L(z1:d/2)` (yellow) - `L(z1:d)` (gray) - **Aggregated Loss**: Final loss `L(z)` (circle with cross). - **Agent/Model**: Stylized figure (red base, yellow top) at the bottom, likely representing the model being trained. --- ### Detailed Analysis - **Inference Flow**: - Shortlisting → Re-ranking → Adaptive Retrieval → Adaptive Classification. - Adaptive Classification outputs a bar chart with diminishing segment sizes (red > blue > yellow), suggesting confidence scores or feature weights. - **Training Hierarchy**: - Loss functions are labeled with decreasing fractions (`1/16` to `1/d`), implying multi-scale or hierarchical training. - The circular `L(z)` with a cross may represent a loss aggregation or optimization step (e.g., gradient descent). - **Connections**: - A dashed arrow links Adaptive Classification to the Training phase, indicating feedback from classification to training losses. - The agent/model at the bottom receives input from the aggregated loss `L(z)`. --- ### Key Observations 1. **Hierarchical Training**: The loss functions (`L(z1:d/16)` to `L(z1:d)`) suggest a multi-resolution training strategy, where finer-grained losses (`1/16`, `1/8`) are prioritized early, and coarser losses (`1/d`) dominate later. 2. **Adaptive Feedback Loop**: The dashed arrow implies that Adaptive Classification’s output influences the training process, enabling dynamic adjustment of the model. 3. **Symbolism**: The circular `L(z)` with a cross may symbolize a loss minimization objective, while the agent/model’s design (red base, yellow top) could represent stability (red) and adaptability (yellow). --- ### Interpretation This diagram represents a **self-improving system** where inference and training are tightly coupled. The hierarchical loss structure in training likely enables the model to learn at multiple scales, while the adaptive classification feedback loop ensures the model refines its predictions iteratively. The stylized agent at the bottom symbolizes the model’s evolving capabilities, shaped by the aggregated loss `L(z)`. The use of color-coded arrows and segmented bars emphasizes modularity and adaptability in both phases. </details> Figure 1: <details> <summary>x5.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ is adaptable to any representation learning setup and begets a ${\rm Matryoshka~Representation}$ $z$ by optimizing the original loss $\mathcal{L}(.)$ at $O(\log(d))$ chosen representation sizes. ${\rm Matryoshka~Representation}$ can be utilized effectively for adaptive deployment across environments and downstream tasks. The first $m$ -dimensions, $m\in[d]$ , of the ${\rm Matryoshka~Representation}$ is an information-rich low-dimensional vector, at no additional training cost, that is as accurate as an independently trained $m$ -dimensional representation. The information within the ${\rm Matryoshka~Representation}$ increases with the dimensionality creating a coarse-to-fine grained representation, all without significant training or additional deployment overhead. ${\rm MRL}$ equips the representation vector with the desired flexibility and multifidelity that can ensure a near-optimal accuracy-vs-compute trade-off. With these advantages, ${\rm MRL}$ enables adaptive deployment based on accuracy and compute constraints. The ${\rm Matryoshka~Representations}$ improve efficiency for large-scale classification and retrieval without any significant loss of accuracy. While there are potentially several applications of coarse-to-fine ${\rm Matryoshka~Representations}$ , in this work we focus on two key building blocks of real-world ML systems: large-scale classification and retrieval. For classification, we use adaptive cascades with the variable-size representations from a model trained with ${\rm MRL}$ , significantly reducing the average dimension of embeddings needed to achieve a particular accuracy. For example, on ImageNet-1K, ${\rm MRL}$ + adaptive classification results in up to a $14\times$ smaller representation size at the same accuracy as baselines (Section 4.2.1). Similarly, we use ${\rm MRL}$ in an adaptive retrieval system. Given a query, we shortlist retrieval candidates using the first few dimensions of the query embedding, and then successively use more dimensions to re-rank the retrieved set. A simple implementation of this approach leads to $128\times$ theoretical (in terms of FLOPS) and $14\times$ wall-clock time speedups compared to a single-shot retrieval system that uses a standard embedding vector; note that ${\rm MRL}$ ’s retrieval accuracy is comparable to that of single-shot retrieval (Section 4.3.1). Finally, as ${\rm MRL}$ explicitly learns coarse-to-fine representation vectors, intuitively it should share more semantic information among its various dimensions (Figure 5). This is reflected in up to $2\$ accuracy gains in long-tail continual learning settings while being as robust as the original embeddings. Furthermore, due to its coarse-to-fine grained nature, ${\rm MRL}$ can also be used as method to analyze hardness of classification among instances and information bottlenecks. We make the following key contributions: 1. We introduce <details> <summary>x6.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) to obtain flexible representations ( ${\rm Matryoshka~Representations}$ ) for adaptive deployment (Section 3). 1. Up to $14\times$ faster yet accurate large-scale classification and retrieval using ${\rm MRL}$ (Section 4). 1. Seamless adaptation of ${\rm MRL}$ across modalities (vision - ResNet & ViT, vision + language - ALIGN, language - BERT) and to web-scale data (ImageNet-1K/4K, JFT-300M and ALIGN data). 1. Further analysis of ${\rm MRL}$ ’s representations in the context of other downstream tasks (Section 5). ## 2 Related Work Representation Learning. Large-scale datasets like ImageNet [16, 76] and JFT [85] enabled the learning of general purpose representations for computer vision [4, 98]. These representations are typically learned through supervised and un/self-supervised learning paradigms. Supervised pretraining [29, 51, 82] casts representation learning as a multi-class/label classification problem, while un/self-supervised learning learns representation via proxy tasks like instance classification [97] and reconstruction [31, 63]. Recent advances [12, 30] in contrastive learning [27] enabled learning from web-scale data [21] that powers large-capacity cross-modal models [18, 46, 71, 101]. Similarly, natural language applications are built [40] on large language models [8] that are pretrained [68, 75] in a un/self-supervised fashion with masked language modelling [19] or autoregressive training [70]. <details> <summary>x7.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) is complementary to all these setups and can be adapted with minimal overhead (Section 3). ${\rm MRL}$ equips representations with multifidelity at no additional cost which enables adaptive deployment based on the data and task (Section 4). Efficient Classification and Retrieval. Efficiency in classification and retrieval during inference can be studied with respect to the high yet constant deep featurization costs or the search cost which scales with the size of the label space and data. Efficient neural networks address the first issue through a variety of algorithms [25, 54] and design choices [39, 53, 87]. However, with a strong featurizer, most of the issues with scale are due to the linear dependence on number of labels ( $L$ ), size of the data ( $N$ ) and representation size ( $d$ ), stressing RAM, disk and processor all at the same time. The sub-linear complexity dependence on number of labels has been well studied in context of compute [3, 43, 69] and memory [20] using Approximate Nearest Neighbor Search (ANNS) [62] or leveraging the underlying hierarchy [17, 55]. In case of the representation size, often dimensionality reduction [77, 88], hashing techniques [14, 52, 78] and feature selection [64] help in alleviating selective aspects of the $O(d)$ scaling at a cost of significant drops in accuracy. Lastly, most real-world search systems [11, 15] are often powered by large-scale embedding based retrieval [10, 66] that scales in cost with the ever increasing web-data. While categorization [89, 99] clusters similar things together, it is imperative to be equipped with retrieval capabilities that can bring forward every instance [7]. Approximate Nearest Neighbor Search (ANNS) [42] makes it feasible with efficient indexing [14] and traversal [5, 6] to present the users with the most similar documents/images from the database for a requested query. Widely adopted HNSW [62] ( $O(d\log(N))$ ) is as accurate as exact retrieval ( $O(dN)$ ) at the cost of a graph-based index overhead for RAM and disk [44]. ${\rm MRL}$ tackles the linear dependence on embedding size, $d$ , by learning multifidelity ${\rm Matryoshka~Representations}$ . Lower-dimensional ${\rm Matryoshka~Representations}$ are as accurate as independently trained counterparts without the multiple expensive forward passes. ${\rm Matryoshka~Representations}$ provide an intermediate abstraction between high-dimensional vectors and their efficient ANNS indices through the adaptive embeddings nested within the original representation vector (Section 4). All other aforementioned efficiency techniques are complementary and can be readily applied to the learned ${\rm Matryoshka~Representations}$ obtained from ${\rm MRL}$ . Several works in efficient neural network literature [9, 93, 100] aim at packing neural networks of varying capacity within the same larger network. However, the weights for each progressively smaller network can be different and often require distinct forward passes to isolate the final representations. This is detrimental for adaptive inference due to the need for re-encoding the entire retrieval database with expensive sub-net forward passes of varying capacities. Several works [23, 26, 65, 59] investigate the notions of intrinsic dimensionality and redundancy of representations and objective spaces pointing to minimum description length [74]. Finally, ordered representations proposed by Rippel et al. [73] use nested dropout in the context of autoencoders to learn nested representations. ${\rm MRL}$ differentiates itself in formulation by optimizing only for $O(\log(d))$ nesting dimensions instead of $O(d)$ . Despite this, ${\rm MRL}$ diffuses information to intermediate dimensions interpolating between the optimized ${\rm Matryoshka~Representation}$ sizes accurately (Figure 5); making web-scale feasible. ## 3 <details> <summary>x8.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ For $d\in\mathbb{N}$ , consider a set $\mathcal{M}\subset[d]$ of representation sizes. For a datapoint $x$ in the input domain $\mathcal{X}$ , our goal is to learn a $d$ -dimensional representation vector $z\in\mathbb{R}^{d}$ . For every $m\in\mathcal{M}$ , ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) enables each of the first $m$ dimensions of the embedding vector, $z_{1:m}\in\mathbb{R}^{m}$ to be independently capable of being a transferable and general purpose representation of the datapoint $x$ . We obtain $z$ using a deep neural network $F(\,\cdot\,;\theta_{F})\colon\mathcal{X}\rightarrow\mathbb{R}^{d}$ parameterized by learnable weights $\theta_{F}$ , i.e., $z\coloneqq F(x;\theta_{F})$ . The multi-granularity is captured through the set of the chosen dimensions $\mathcal{M}$ , that contains less than $\log(d)$ elements, i.e., $\lvert\mathcal{M}\rvert\leq\left\lfloor\log(d)\right\rfloor$ . The usual set $\mathcal{M}$ consists of consistent halving until the representation size hits a low information bottleneck. We discuss the design choices in Section 4 for each of the representation learning settings. For the ease of exposition, we present the formulation for fully supervised representation learning via multi-class classification. ${\rm Matryoshka~Representation~Learning}$ modifies the typical setting to become a multi-scale representation learning problem on the same task. For example, we train ResNet50 [29] on ImageNet-1K [76] which embeds a $224\times 224$ pixel image into a $d=2048$ representation vector and then passed through a linear classifier to make a prediction, $\hat{y}$ among the $L=1000$ labels. For ${\rm MRL}$ , we choose $\mathcal{M}=\{8,16,\ldots,1024,2048\}$ as the nesting dimensions. Suppose we are given a labelled dataset $\mathcal{D}=\{(x_{1},y_{1}),\ldots,(x_{N},y_{N})\}$ where $x_{i}\in\mathcal{X}$ is an input point and $y_{i}\in[L]$ is the label of $x_{i}$ for all $i\in[N]$ . ${\rm MRL}$ optimizes the multi-class classification loss for each of the nested dimension $m\in\mathcal{M}$ using standard empirical risk minimization using a separate linear classifier, parameterized by $\mathbf{W}^{(m)}\in\mathbb{R}^{L\times m}$ . All the losses are aggregated after scaling with their relative importance $\left(c_{m}\geq 0\right)_{m\in\mathcal{M}}$ respectively. That is, we solve $$ \min_{\left\{\mathbf{W}^{(m)}\right\}_{m\in\mathcal{M}},\ \theta_{F}}\frac{1}{N}\sum_{i\in[N]}\sum_{m\in\mathcal{M}}c_{m}\cdot{\cal L}\left(\mathbf{W}^{(m)}\cdot F(x_{i};\theta_{F})_{1:m}\ ;\ y_{i}\right)\ , \tag{1} $$ where ${\cal L}\colon\mathbb{R}^{L}\times[L]\to\mathbb{R}_{+}$ is the multi-class softmax cross-entropy loss function. This is a standard optimization problem that can be solved using sub-gradient descent methods. We set all the importance scales, $c_{m}=1$ for all $m\in\mathcal{M}$ ; see Section 5 for ablations. Lastly, despite only optimizing for $O(\log(d))$ nested dimensions, ${\rm MRL}$ results in accurate representations, that interpolate, for dimensions that fall between the chosen granularity of the representations (Section 4.2). We call this formulation as ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ). A natural way to make this efficient is through weight-tying across all the linear classifiers, i.e., by defining $\mathbf{W}^{(m)}=\mathbf{W}_{1:m}$ for a set of common weights $\mathbf{W}\in\mathbb{R}^{L\times d}$ . This would reduce the memory cost due to the linear classifiers by almost half, which would be crucial in cases of extremely large output spaces [89, 99]. This variant is called Efficient ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL\text{--}E}$ ). Refer to Alg 1 and Alg 2 in Appendix A for the building blocks of ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ). Adaptation to Learning Frameworks. ${\rm MRL}$ can be adapted seamlessly to most representation learning frameworks at web-scale with minimal modifications (Section 4.1). For example, ${\rm MRL}$ ’s adaptation to masked language modelling reduces to ${\rm MRL\text{--}E}$ due to the weight-tying between the input embedding matrix and the linear classifier. For contrastive learning, both in context of vision & vision + language, ${\rm MRL}$ is applied to both the embeddings that are being contrasted with each other. The presence of normalization on the representation needs to be handled independently for each of the nesting dimension for best results (see Appendix C for more details). ## 4 Applications In this section, we discuss ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) for a diverse set of applications along with an extensive evaluation of the learned multifidelity representations. Further, we showcase the downstream applications of the learned ${\rm Matryoshka~Representations}$ for flexible large-scale deployment through (a) Adaptive Classification (AC) and (b) Adaptive Retrieval (AR). <details> <summary>x9.png Details</summary>  ### Visual Description ## Line Graph: Top-1 Accuracy vs. Representation Size ### Overview The image is a line graph comparing the Top-1 Accuracy (%) of six different methods (MRL, MRL-E, FF, SVD, Slim. Net, Rand. LP) across varying Representation Sizes (8 to 2048). The graph shows how accuracy improves or plateaus as representation size increases. ### Components/Axes - **X-axis (Horizontal)**: Representation Size (logarithmic scale: 8, 16, 32, 64, 128, 256, 512, 1024, 2048). - **Y-axis (Vertical)**: Top-1 Accuracy (%) (linear scale: 40% to 80%). - **Legend**: Located in the top-right corner, mapping colors/symbols to methods: - **Blue (solid line with circles)**: MRL - **Orange (dashed line with triangles)**: MRL-E - **Green (dotted line with triangles)**: FF - **Red (dash-dot line with hexagons)**: SVD - **Purple (dash-dot line with crosses)**: Slim. Net - **Brown (solid line with crosses)**: Rand. LP ### Detailed Analysis 1. **MRL (Blue)**: Starts at ~65% accuracy at size 8, rises steadily to ~75% by size 16, and plateaus near 75% for larger sizes. 2. **MRL-E (Orange)**: Begins at ~55% at size 8, increases sharply to ~70% by size 16, and plateaus near 70% for larger sizes. 3. **FF (Green)**: Starts at ~60% at size 8, rises to ~70% by size 16, and plateaus near 70% for larger sizes. 4. **SVD (Red)**: Begins at ~50% at size 8, rises steeply to ~70% by size 128, and plateaus near 70% for larger sizes. 5. **Slim. Net (Purple)**: Starts at ~45% at size 8, increases gradually to ~70% by size 1024, and plateaus near 70% for larger sizes. 6. **Rand. LP (Brown)**: Starts at ~40% at size 8, rises sharply to ~70% by size 1024, and plateaus near 70% for larger sizes. ### Key Observations - **Performance Trends**: All methods improve accuracy with larger representation sizes, but MRL and MRL-E achieve the highest initial accuracy and plateau earlier. - **Steepest Growth**: Rand. LP and SVD show the most significant improvement as representation size increases, though they start from lower baselines. - **Plateaus**: Most methods plateau near 70–75% accuracy, suggesting diminishing returns at larger sizes. - **Outliers**: Slim. Net and Rand. LP lag behind others initially but catch up at larger sizes. ### Interpretation The graph demonstrates that **MRL and MRL-E** are the most efficient methods, achieving high accuracy even at smaller representation sizes. In contrast, **Rand. LP** and **SVD** require larger representations to reach comparable performance, indicating they may be less efficient but potentially more scalable. The plateauing trends suggest that increasing representation size beyond a certain point yields minimal accuracy gains for most methods. This could inform trade-offs between computational cost and performance in practical applications. </details> Figure 2: ImageNet-1K linear classification accuracy of ResNet50 models. ${\rm MRL}$ is as accurate as the independently trained FF models for every representation size. <details> <summary>x10.png Details</summary>  ### Visual Description ## Line Graph: 1-NN Accuracy vs. Representation Size ### Overview The image is a line graph comparing the 1-NN (1-Nearest Neighbor) accuracy of various algorithms as a function of representation size. The x-axis represents representation size (in powers of 2: 8, 16, 32, ..., 2048), and the y-axis represents accuracy percentage (40% to 70%). Six algorithms are compared: MRL, MRL-E, FF, SVD, Slim. Net, and Rand. FS. Each algorithm is represented by a distinct line style and color. --- ### Components/Axes - **X-axis (Representation Size)**: - Labels: 8, 16, 32, 64, 128, 256, 512, 1024, 2048. - Scale: Logarithmic (powers of 2). - **Y-axis (1-NN Accuracy (%))**: - Labels: 40%, 50%, 60%, 70%. - Scale: Linear (0% to 70%). - **Legend**: - Position: Right side of the graph. - Entries: - **MRL**: Solid blue line with circles. - **MRL-E**: Dashed orange line with triangles. - **FF**: Dotted green line with triangles. - **SVD**: Dash-dot red line with hexagons. - **Slim. Net**: Dash-dot purple line with crosses. - **Rand. FS**: Solid brown line with crosses. --- ### Detailed Analysis 1. **MRL (Solid Blue)**: - Starts at ~62% accuracy at 8x representation size. - Increases steadily to ~70% by 32x, then plateaus. - Maintains ~70% accuracy for larger sizes (64x–2048x). 2. **MRL-E (Dashed Orange)**: - Starts at ~58% at 8x. - Rises to ~68% by 32x, then plateaus. - Slightly lower than MRL for all sizes. 3. **FF (Dotted Green)**: - Starts at ~59% at 8x. - Increases to ~69% by 32x, then plateaus. - Similar performance to MRL but slightly lower. 4. **SVD (Dash-Dot Red)**: - Starts at ~45% at 8x. - Sharp increase to ~65% by 32x, then plateaus. - Highest initial growth rate but lower plateau than MRL/FF. 5. **Slim. Net (Dash-Dot Purple)**: - Starts at ~55% at 8x. - Increases to ~67% by 32x, then plateaus. - Moderate growth rate and lower plateau than MRL/FF. 6. **Rand. FS (Solid Brown)**: - Starts at ~50% at 8x. - Rapid increase to ~70% by 32x, then plateaus. - Matches MRL/FF performance at larger sizes despite slow start. --- ### Key Observations - **Trend Verification**: - All algorithms show **increasing accuracy** with larger representation sizes, but the rate of improvement varies. - **SVD** and **Rand. FS** exhibit the steepest initial growth, while **MRL** and **FF** achieve the highest plateau. - **MRL-E** and **Slim. Net** have moderate growth and lower plateaus. - **Notable Patterns**: - **MRL** and **FF** consistently outperform others at larger sizes. - **SVD** and **Rand. FS** show strong scalability but require larger representation sizes to reach peak performance. - **Slim. Net** and **MRL-E** lag behind in both growth and final accuracy. --- ### Interpretation The data suggests that **MRL** and **FF** are the most effective algorithms for 1-NN tasks, achieving high accuracy even at smaller representation sizes. **SVD** and **Rand. FS** demonstrate strong scalability but require larger data representations to match the performance of MRL/FF. The **Slim. Net** and **MRL-E** algorithms show moderate performance, indicating potential inefficiencies in their design or training. The sharp initial growth of **SVD** and **Rand. FS** implies they may be particularly sensitive to representation size, possibly due to their reliance on specific features or dimensionality reduction techniques. Overall, the graph highlights the trade-off between representation size and algorithmic efficiency in 1-NN tasks. </details> Figure 3: ImageNet-1K 1-NN accuracy of ResNet50 models measuring the representation quality for downstream task. ${\rm MRL}$ outperforms all the baselines across all representation sizes. ### 4.1 Representation Learning We adapt ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) to various representation learning setups (a) Supervised learning for vision: ResNet50 [29] on ImageNet-1K [76] and ViT-B/16 [22] on JFT-300M [85], (b) Contrastive learning for vision + language: ALIGN model with ViT-B/16 vision encoder and BERT language encoder on ALIGN data [46] and (c) Masked language modelling: BERT [19] on English Wikipedia and BooksCorpus [102]. Please refer to Appendices B and C for details regarding the model architectures, datasets and training specifics. We do not search for best hyper-parameters for all ${\rm MRL}$ experiments but use the same hyper-parameters as the independently trained baselines. ResNet50 outputs a $2048$ -dimensional representation while ViT-B/16 and BERT-Base output $768$ -dimensional embeddings for each data point. We use $\mathcal{M}=\{8,16,32,64,128,256,512,1024,2048\}$ and $\mathcal{M}=\{12,24,48,96,192,384,768\}$ as the explicitly optimized nested dimensions respectively. Lastly, we extensively compare the ${\rm MRL}$ and ${\rm MRL\text{--}E}$ models to independently trained low-dimensional (fixed feature) representations (FF), dimensionality reduction (SVD), sub-net method (slimmable networks [100]) and randomly selected features of the highest capacity FF model. In section 4.2, we evaluate the quality and capacity of the learned representations through linear classification/probe (LP) and 1-nearest neighbour (1-NN) accuracy. Experiments show that ${\rm MRL}$ models remove the dependence on $|\mathcal{M}|$ resource-intensive independently trained models for the coarse-to-fine representations while being as accurate. Lastly, we show that despite optimizing only for $|\mathcal{M}|$ dimensions, ${\rm MRL}$ models diffuse the information, in an interpolative fashion, across all the $d$ dimensions providing the finest granularity required for adaptive deployment. ### 4.2 Classification Figure 3 compares the linear classification accuracy of ResNet50 models trained and evaluated on ImageNet-1K. ResNet50– ${\rm MRL}$ model is at least as accurate as each FF model at every representation size in $\mathcal{M}$ while ${\rm MRL\text{--}E}$ is within $1\$ starting from $16$ -dim. Similarly, Figure 3 showcases the comparison of learned representation quality through 1-NN accuracy on ImageNet-1K (trainset with 1.3M samples as the database and validation set with 50K samples as the queries). ${\rm Matryoshka~Representations}$ are up to $2\$ more accurate than their fixed-feature counterparts for the lower-dimensions while being as accurate elsewhere. 1-NN accuracy is an excellent proxy, at no additional training cost, to gauge the utility of learned representations in the downstream tasks. We also evaluate the quality of the representations from training ViT-B/16 on JFT-300M alongside the ViT-B/16 vision encoder of the ALIGN model – two web-scale setups. Due to the expensive nature of these experiments, we only train the highest capacity fixed feature model and choose random features for evaluation in lower-dimensions. Web-scale is a compelling setting for ${\rm MRL}$ due to its relatively inexpensive training overhead while providing multifidelity representations for downstream tasks. Figure 5, evaluated with 1-NN on ImageNet-1K, shows that all the ${\rm MRL}$ models for JFT and ALIGN are highly accurate while providing an excellent cost-vs-accuracy trade-off at lower-dimensions. These experiments show that ${\rm MRL}$ seamlessly scales to large-scale models and web-scale datasets while providing the otherwise prohibitively expensive multi-granularity in the process. We also have similar observations when pretraining BERT; please see Appendix D.2 for more details. <details> <summary>x11.png Details</summary>  ### Visual Description ## Line Graph: 1-NN Accuracy vs. Representation Size ### Overview The image is a line graph comparing the 1-NN (1-Nearest Neighbor) classification accuracy of five different methods across varying representation sizes. The x-axis represents representation size (logarithmically scaled), and the y-axis shows accuracy percentage. Five data series are plotted with distinct line styles and markers, each corresponding to a method in the legend. ### Components/Axes - **X-axis (Representation Size)**: Logarithmic scale with markers at 12, 24, 48, 96, 192, 384, and 768. - **Y-axis (1-NN Accuracy %)**: Linear scale from 0% to 80% in 20% increments. - **Legend**: Located in the top-right corner, with five entries: - **JFT MRL**: Solid blue line with circle markers. - **ALIGN MRL**: Dashed orange line with triangle markers. - **JFT MRL-E**: Dotted green line with arrow markers. - **JFT Rand.**: Dotted red line with hexagon markers. - **ALIGN Rand.**: Dashed purple line with plus markers. ### Detailed Analysis 1. **JFT MRL (Blue)**: - Starts at ~50% accuracy at 12 representation size. - Increases steadily to ~70% at 768. - Slope is consistent, with minimal fluctuation. 2. **ALIGN MRL (Orange)**: - Begins at ~40% at 12, rising to ~65% at 768. - Slope is slightly steeper than JFT MRL but plateaus near 65% after 192. 3. **JFT MRL-E (Green)**: - Similar trend to JFT MRL but consistently ~5% lower. - Starts at ~45% at 12, reaching ~65% at 768. 4. **JFT Rand. (Red)**: - Starts at ~25% at 12, rising sharply to ~60% at 768. - Slope is steeper than JFT MRL-E but less consistent. 5. **ALIGN Rand. (Purple)**: - Begins at ~10% at 12, increasing to ~60% at 768. - Slope is the steepest among all series but shows variability. ### Key Observations - **Performance Trends**: All methods improve with larger representation sizes, but the rate of improvement diminishes as size increases. - **JFT MRL vs. ALIGN MRL**: JFT MRL consistently outperforms ALIGN MRL across all sizes, though the gap narrows at larger sizes. - **Random Baselines**: Both JFT Rand. and ALIGN Rand. start significantly lower but converge with non-random methods at larger sizes (~60% at 768). - **JFT MRL-E**: Underperforms JFT MRL but outperforms all random methods. ### Interpretation The data suggests that **JFT MRL** is the most robust method, maintaining high accuracy across representation sizes. **ALIGN MRL** and **JFT MRL-E** show comparable performance to JFT MRL but with slight trade-offs in consistency or magnitude. The random methods (JFT Rand. and ALIGN Rand.) demonstrate that structured approaches (MRL) significantly outperform random baselines, especially at smaller representation sizes. The convergence of random methods at larger sizes implies diminishing returns for structured methods as representation complexity increases. This could indicate that larger representations inherently capture more discriminative features, reducing the need for sophisticated alignment or error-correction mechanisms. </details> Figure 4: ImageNet-1K 1-NN accuracy for ViT-B/16 models trained on JFT-300M & as part of ALIGN. ${\rm MRL}$ scales seamlessly to web-scale with minimal training overhead. <details> <summary>x12.png Details</summary>  ### Visual Description ## Line Graph: 1-NN Accuracy vs. Representation Size ### Overview The image is a line graph comparing the 1-NN accuracy of different models across varying representation sizes. The x-axis represents representation size (8 to 2048), and the y-axis shows accuracy percentage (50% to 70%). Six data series are plotted with distinct line styles and markers, each corresponding to a model or variant. ### Components/Axes - **X-axis (Representation Size)**: Logarithmic scale from 8 to 2048 (values: 8, 16, 32, 64, 128, 256, 512, 1024, 2048). - **Y-axis (1-NN Accuracy %)**: Linear scale from 50% to 70%. - **Legend**: Located in the bottom-right corner, mapping colors/markers to models: - **Green dashed line with triangles**: ViT-ALIGN - **Blue dashed line with circles**: ViT-JFT - **Purple dashed line with squares**: RN50-IN1K - **Green dashed line with triangles**: ViT-ALIGN-Int - **Blue dashed line with circles**: ViT-JFT-Int - **Purple dashed line with squares**: RN50-IN1K-Int ### Detailed Analysis 1. **ViT-ALIGN** (green dashed line with triangles): - Starts at ~50% accuracy at 8 representation size. - Gradually increases to ~65% at 2048. - Slowest growth among all series. 2. **ViT-JFT** (blue dashed line with circles): - Begins at ~55% at 8. - Rises sharply to ~70% by 128. - Plateaus near 70% for larger sizes. 3. **RN50-IN1K** (purple dashed line with squares): - Starts at ~60% at 8. - Increases to ~70% by 32. - Remains stable at ~70% for larger sizes. 4. **ViT-ALIGN-Int** (green dashed line with triangles): - Identical trend to ViT-ALIGN (same color/marker). - Starts at ~50% and reaches ~65% at 2048. 5. **ViT-JFT-Int** (blue dashed line with circles): - Matches ViT-JFT trend (same color/marker). - Starts at ~55% and plateaus at ~70%. 6. **RN50-IN1K-Int** (purple dashed line with squares): - Mirrors RN50-IN1K (same color/marker). - Starts at ~60% and stabilizes at ~70%. ### Key Observations - **Trend Consistency**: All models show increasing accuracy with larger representation sizes, but the rate of improvement varies. - **Performance Hierarchy**: RN50-IN1K variants outperform ViT models, which outperform ViT-ALIGN variants. - **Legend Ambiguity**: The "Int" variants (e.g., ViT-ALIGN-Int) share identical line styles and trends with their non-"Int" counterparts, suggesting potential labeling errors or redundancy. - **Plateau Effect**: RN50-IN1K models plateau earlier (by 32) compared to ViT-JFT (128) and ViT-ALIGN (2048). ### Interpretation The data demonstrates that larger representation sizes improve 1-NN accuracy across all models, with RN50-IN1K achieving the highest performance. The "Int" variants appear to replicate the trends of their base models, raising questions about their distinctiveness. The steepest growth is observed in ViT-JFT, suggesting it benefits most from increased representation size. The RN50-IN1K models’ early plateau indicates diminishing returns at smaller sizes, while ViT-ALIGN’s gradual improvement highlights its sensitivity to representation scale. The legend’s duplication of styles for "Int" variants warrants verification, as it may obscure meaningful distinctions between model configurations. </details> Figure 5: Despite optimizing ${\rm MRL}$ only for $O(\log(d))$ dimensions for ResNet50 and ViT-B/16 models; the accuracy in the intermediate dimensions shows interpolating behaviour. Our experiments also show that post-hoc compression (SVD), linear probe on random features, and sub-net style slimmable networks drastically lose accuracy compared to ${\rm MRL}$ as the representation size decreases. Finally, Figure 5 shows that, while ${\rm MRL}$ explicitly optimizes $O(\log(d))$ nested representations – removing the $O(d)$ dependence [73] –, the coarse-to-fine grained information is interpolated across all $d$ dimensions providing highest flexibility for adaptive deployment. #### 4.2.1 Adaptive Classification The flexibility and coarse-to-fine granularity within ${\rm Matryoshka~Representations}$ allows model cascades [90] for Adaptive Classification (AC) [28]. Unlike standard model cascades [95], ${\rm MRL}$ does not require multiple expensive neural network forward passes. To perform AC with an ${\rm MRL}$ trained model, we learn thresholds on the maximum softmax probability [33] for each nested classifier on a holdout validation set. We then use these thresholds to decide when to transition to the higher dimensional representation (e.g $8\to 16\to 32$ ) of the ${\rm MRL}$ model. Appendix D.1 discusses the implementation and learning of thresholds for cascades used for adaptive classification in detail. Figure 7 shows the comparison between cascaded ${\rm MRL}$ representations ( ${\rm MRL}$ –AC) and independently trained fixed feature (FF) models on ImageNet-1K with ResNet50. We computed the expected representation size for ${\rm MRL}$ –AC based on the final dimensionality used in the cascade. We observed that ${\rm MRL}$ –AC was as accurate, $76.30\$ , as a 512-dimensional FF model but required an expected dimensionality of $\sim 37$ while being only $0.8\$ lower than the 2048-dimensional FF baseline. Note that all ${\rm MRL}$ –AC models are significantly more accurate than the FF baselines at comparable representation sizes. ${\rm MRL}$ –AC uses up to $\sim 14\times$ smaller representation size for the same accuracy which affords computational efficiency as the label space grows [89]. Lastly, our results with ${\rm MRL}$ –AC indicate that instances and classes vary in difficulty which we analyze in Section 5 and Appendix J. ### 4.3 Retrieval Nearest neighbour search with learned representations powers a plethora of retrieval and search applications [15, 91, 11, 66]. In this section, we discuss the image retrieval performance of the pretrained ResNet50 models (Section 4.1) on two large-scale datasets ImageNet-1K [76] and ImageNet-4K. ImageNet-1K has a database size of $\sim$ 1.3M and a query set of 50K samples uniformly spanning 1000 classes. We also introduce ImageNet-4K which has a database size of $\sim$ 4.2M and query set of $\sim$ 200K samples uniformly spanning 4202 classes (see Appendix B for details). A single forward pass on ResNet50 costs 4 GFLOPs while exact retrieval costs 2.6 GFLOPs per query for ImageNet-1K. Although retrieval overhead is $40\$ of the total cost, retrieval cost grows linearly with the size of the database. ImageNet-4K presents a retrieval benchmark where the exact search cost becomes the computational bottleneck ( $8.6$ GFLOPs per query). In both these settings, the memory and disk usage are also often bottlenecked by the large databases. However, in most real-world applications exact search, $O(dN)$ , is replaced with an approximate nearest neighbor search (ANNS) method like HNSW [62], $O(d\log(N))$ , with minimal accuracy drop at the cost of additional memory overhead. The goal of image retrieval is to find images that belong to the same class as the query using representations obtained from a pretrained model. In this section, we compare retrieval performance using mean Average Precision @ 10 (mAP@ $10$ ) which comprehensively captures the setup of relevant image retrieval at scale. We measure the cost per query using exact search in MFLOPs. All embeddings are unit normalized and retrieved using the L2 distance metric. Lastly, we report an extensive set of metrics spanning mAP@ $k$ and P@ $k$ for $k=\{10,25,50,100\}$ and real-world wall-clock times for exact search and HNSW. See Appendices E and F for more details. <details> <summary>x13.png Details</summary>  ### Visual Description ## Line Chart: Top-1 Accuracy vs. Representation Size ### Overview The chart compares the Top-1 Accuracy (%) of two models (MRL-AC and FF) across varying representation sizes, with annotations for a baseline (FF 2048) and a reference point ("14x smaller representation size"). The x-axis represents representation size, and the y-axis shows accuracy. ### Components/Axes - **X-axis (Horizontal)**: "(Expected) Representation Size" with values: 16, 32, 64, 128, 256, 512. - **Y-axis (Vertical)**: "Top-1 Accuracy (%)" with values from 74% to 77%. - **Legend**: Located in the bottom-right corner, associating: - Blue circles: MRL-AC - Orange crosses: FF - Purple dashed line: FF 2048 - **Annotations**: - Green dashed line labeled "14x smaller representation size" at x=32. - Purple dashed line labeled "FF 2048" at y=77%. ### Detailed Analysis - **MRL-AC (Blue Circles)**: - Data points: (16, 75.2%), (32, 76.1%), (64, 76.3%), (128, 76.4%), (256, 76.5%), (512, 76.6%). - Trend: Slight upward trajectory as representation size increases. - **FF (Orange Crosses)**: - Data points: (32, 74.8%), (64, 75.3%), (128, 75.5%), (256, 75.7%), (512, 75.9%). - Trend: Gradual upward trend but consistently below MRL-AC. - **FF 2048 (Purple Dashed Line)**: Horizontal line at 77%, above all data points. - **Green Dashed Line**: Vertical reference at x=32, labeled "14x smaller representation size." ### Key Observations 1. **MRL-AC outperforms FF** across all representation sizes, with a maximum accuracy of 76.6% vs. FF's 75.9%. 2. **FF 2048 baseline** (77%) is unattained by either model, suggesting it represents an ideal or theoretical limit. 3. **14x smaller representation size** at x=32 aligns with MRL-AC's 76.1% accuracy, indicating efficiency at reduced scale. 4. **FF's performance** improves with larger representation sizes but remains inferior to MRL-AC. ### Interpretation The data demonstrates that MRL-AC achieves higher accuracy than FF across all tested representation sizes, with a consistent gap of ~0.3–0.5%. The FF 2048 line (77%) acts as a ceiling, implying potential for further optimization. The "14x smaller representation size" annotation at x=32 highlights MRL-AC's efficiency, maintaining strong performance even at reduced scale. This suggests MRL-AC may be more robust or optimized for resource-constrained scenarios compared to FF. </details> Figure 6: Adaptive classification on ${\rm MRL}$ ResNet50 using cascades results in $14\times$ smaller representation size for the same level of accuracy on ImageNet-1K ( $\sim 37$ vs $512$ dims for $76.3\$ ). <details> <summary>x14.png Details</summary>  ### Visual Description ## Line Graph: mAP@10% Performance vs. Representation Size ### Overview The graph compares the mean Average Precision at 10% (mAP@10%) performance of six different methods across varying representation sizes (8 to 2048). The y-axis ranges from 40% to 65%, while the x-axis uses logarithmic scaling (powers of 2). All lines show distinct trends, with some plateauing early and others improving significantly with larger representations. ### Components/Axes - **X-axis (Representation Size)**: Logarithmic scale from 8 to 2048 (steps: 8, 16, 32, 64, 128, 256, 512, 1024, 2048). - **Y-axis (mAP@10%)**: Linear scale from 40% to 65%. - **Legend**: Located in the top-right corner, mapping colors and line styles to methods: - **MRL**: Blue solid line with circles. - **MRL-E**: Orange dashed line with triangles. - **FF**: Green dash-dot line with downward triangles. - **SVD**: Red dotted line with hexagons. - **Slim. Net**: Purple dash-dot line with crosses. - **Rand. FS**: Brown solid line with crosses. ### Detailed Analysis 1. **MRL (Blue)**: Starts at ~57% at 8, rises sharply to ~64% by 16, then plateaus. Maintains ~64% across all larger sizes. 2. **MRL-E (Orange)**: Begins at ~52% at 8, increases to ~63% by 16, then plateaus. Slightly below MRL but stable. 3. **FF (Green)**: Starts at ~54% at 8, peaks at ~63% at 16, then dips slightly (~62%) by 2048. 4. **SVD (Red)**: Begins at ~50% at 8, rises to ~62% by 128, then dips to ~60% by 2048. 5. **Slim. Net (Purple)**: Starts at ~53% at 8, peaks at ~61% at 128, then declines to ~58% by 2048. 6. **Rand. FS (Brown)**: Starts at ~40% at 8, rises sharply to ~63% by 2048, showing the steepest improvement. ### Key Observations - **Early Performance**: MRL and MRL-E dominate at smaller representation sizes (8–16), achieving ~64% mAP@10%. - **Scalability**: Rand. FS improves dramatically with larger representations, suggesting it benefits from increased data. - **Dips**: SVD and Slim. Net show performance declines after 128, possibly due to overfitting or inefficiency at higher sizes. - **Consistency**: MRL and MRL-E maintain stable performance across all sizes, indicating robustness. ### Interpretation The data suggests that **MRL and MRL-E** are the most efficient methods for small-to-medium representation sizes, while **Rand. FS** excels with larger datasets. The decline in SVD and Slim. Net at higher sizes may indicate suboptimal scaling or overfitting. The logarithmic x-axis highlights that performance gains are most pronounced in the early stages for most methods, except Rand. FS, which shows linear improvement. This could imply that Rand. FS is better suited for high-dimensional data, whereas MRL/MRL-E are optimized for compact representations. </details> Figure 7: mAP@ $10$ for Image Retrieval on ImageNet-1K with ResNet50. ${\rm MRL}$ consistently produces better retrieval performance over the baselines across all the representation sizes. Figure 7 compares the mAP@ $10$ performance of ResNet50 representations on ImageNet-1K across dimensionalities for ${\rm MRL}$ , ${\rm MRL\text{--}E}$ , FF, slimmable networks along with post-hoc compression of vectors using SVD and random feature selection. ${\rm Matryoshka~Representations}$ are often the most accurate while being up to $3\$ better than the FF baselines. Similar to classification, post-hoc compression and slimmable network baselines suffer from significant drop-off in retrieval mAP@ $10$ with $\leq 256$ dimensions. Appendix E discusses the mAP@ $10$ of the same models on ImageNet-4K. ${\rm MRL}$ models are capable of performing accurate retrieval at various granularities without the additional expense of multiple model forward passes for the web-scale databases. FF models also generate independent databases which become prohibitively expense to store and switch in between. ${\rm Matryoshka~Representations}$ enable adaptive retrieval (AR) which alleviates the need to use full-capacity representations, $d=2048$ , for all data and downstream tasks. Lastly, all the vector compression techniques [60, 45] used as part of the ANNS pipelines are complimentary to ${\rm Matryoshka~Representations}$ and can further improve the efficiency-vs-accuracy trade-off. #### 4.3.1 Adaptive Retrieval We benchmark ${\rm MRL}$ in the adaptive retrieval setting (AR) [50]. For a given query image, we obtained a shortlist, $K=200$ , of images from the database using a lower-dimensional representation, e.g. $D_{s}=16$ followed by reranking with a higher capacity representation, e.g. $D_{r}=2048$ . In real-world scenarios where top ranking performance is the key objective, measured with mAP@ $k$ where k covers a limited yet crucial real-estate, AR provides significant compute and memory gains over single-shot retrieval with representations of fixed dimensionality. Finally, the most expensive part of AR, as with any retrieval pipeline, is the nearest neighbour search for shortlisting. For example, even naive re-ranking of 200 images with 2048 dimensions only costs 400 KFLOPs. While we report exact search cost per query for all AR experiments, the shortlisting component of the pipeline can be sped-up using ANNS (HNSW). Appendix I has a detailed discussion on compute cost for exact search, memory overhead of HNSW indices and wall-clock times for both implementations. We note that using HNSW with 32 neighbours for shortlisting does not decrease accuracy during retrieval. | <details> <summary>x15.png Details</summary>  ### Visual Description ## Scatter Plot: mAP@10 vs. MFLOPS/Query with Speed-Up Trends ### Overview The image is a scatter plot comparing **mAP@10 (%)** (mean Average Precision at 10 results) against **MFLOPS/Query** (millions of floating-point operations per query). It includes two trend lines representing theoretical and real-world speed-up factors, along with annotated data points and a "Funnel" marker. --- ### Components/Axes - **X-axis**: "MFLOPS/Query" (logarithmic scale: 10² to 10³). - **Y-axis**: "mAP@10 (%)" (linear scale: 64.9 to 65.3). - **Legend**: - **Red "Y"**: Labeled "Funnel" (positioned at the lower-left cluster of data points). - **Dashed green line**: "128x theoretical speed-up" (top-left to bottom-right). - **Dotted orange line**: "14x real-world speed-up" (bottom-left to top-right). --- ### Detailed Analysis 1. **Data Points**: - **Blue dots**: Clustered at lower MFLOPS/Query values (10² to ~10².5), with mAP@10 ranging from 65.0 to 65.2. - **Purple dots**: Spread across higher MFLOPS/Query values (10².5 to 10³), with mAP@10 between 65.0 and 65.2. - **Red "Y" markers**: Located at the lowest MFLOPS/Query (10²) and highest mAP@10 (65.2), suggesting a focal point for the "Funnel" model. 2. **Trend Lines**: - **Green dashed line ("128x theoretical speed-up")**: - Starts at ~65.3 mAP@10 (10² MFLOPS/Query) and slopes downward to ~65.1 at 10³ MFLOPS/Query. - Indicates a **theoretical degradation** in performance as computational power increases. - **Orange dotted line ("14x real-world speed-up")**: - Begins at ~65.0 mAP@10 (10² MFLOPS/Query) and rises to ~65.2 at 10³ MFLOPS/Query. - Shows a **real-world improvement** in performance with higher MFLOPS/Query. 3. **Annotations**: - A green arrow points to the highest mAP@10 value (65.3) at the lowest MFLOPS/Query (10²), emphasizing the theoretical peak. - The red "Y" marker is explicitly labeled "Funnel," likely representing a specific optimization or benchmark. --- ### Key Observations - **Theoretical vs. Real-World Divergence**: The green line (theoretical) predicts a **128x speed-up** but shows a **decline in mAP@10** as MFLOPS/Query increases, while the orange line (real-world) demonstrates a **14x speed-up** with **improving mAP@10**. - **Funnel Marker**: The red "Y" at (10² MFLOPS/Query, 65.2 mAP@10) may represent an optimal balance between computational efficiency and performance. - **Data Point Distribution**: - Lower MFLOPS/Query values (10²) cluster around higher mAP@10 (65.2–65.3). - Higher MFLOPS/Query values (10³) show a mix of mAP@10 values, suggesting diminishing returns or trade-offs. --- ### Interpretation The chart highlights a **discrepancy between theoretical and real-world performance gains**. While the "128x theoretical speed-up" line assumes ideal conditions, the "14x real-world speed-up" reflects practical constraints (e.g., algorithmic inefficiencies, hardware limitations). The "Funnel" marker (red "Y") likely signifies a critical threshold where computational resources are optimally allocated to maximize mAP@10. The downward trend of the green line suggests that theoretical models may overestimate performance gains, while the upward orange line underscores the importance of real-world validation. The data implies that increasing MFLOPS/Query does not always linearly improve mAP@10, emphasizing the need for balanced system design. </details> | <details> <summary>x16.png Details</summary>  ### Visual Description ## Scatter Plot: Comparison of D_s and D_r Values ### Overview The image displays a scatter plot comparing two datasets, **D_s** (left column) and **D_r** (right column), with numerical values increasing exponentially from 8 to 2048. Dots represent data points, with size and color gradients corresponding to values. The legend maps colors to specific values, and dot sizes vary between columns. --- ### Components/Axes - **Axes**: - **Left Column (D_s)**: Labeled with values 8, 16, 32, 64, 128, 256, 512, 1024, 2048 (y-axis). - **Right Column (D_r)**: Same numerical values as D_s (y-axis). - **Legend**: - Positioned on the right side of the plot. - Colors range from light gray (8) to dark purple (2048), with intermediate shades for intermediate values. - Each color corresponds to a specific value (e.g., light blue = 16, dark blue = 2048). - **Dot Sizes**: - D_s dots are significantly larger than D_r dots for equivalent values. - No explicit scale for dot size is provided. --- ### Detailed Analysis - **D_s Column**: - Values increase exponentially (powers of 2: 8, 16, 32, ..., 2048). - Dots are large and densely packed, with colors transitioning from light gray (8) to dark purple (2048). - Example: The dot for 2048 is the largest and darkest purple. - **D_r Column**: - Identical numerical values to D_s but represented by smaller dots. - Colors follow the same gradient as D_s but are less saturated due to smaller dot size. - Example: The 2048 value in D_r is a small, dark blue dot. - **Legend Placement**: - Located to the right of both columns, aligned vertically. - Colors are mapped directly to values, with no overlap or ambiguity. --- ### Key Observations 1. **Dot Size Discrepancy**: D_s dots are ~5–10x larger than D_r dots for equivalent values, suggesting a visual emphasis on D_s. 2. **Color Consistency**: Colors in both columns align perfectly with the legend, confirming accurate value representation. 3. **Exponential Scale**: Values double sequentially, indicating a focus on logarithmic or exponential relationships. 4. **No Overlap**: D_s and D_r dots occupy separate vertical columns, preventing direct spatial comparison. --- ### Interpretation - **Purpose**: The plot likely compares two datasets (D_s and D_r) with identical values but differing magnitudes, as indicated by dot size. D_s may represent a primary metric, while D_r could be a scaled or secondary measurement. - **Trend**: Both datasets follow an exponential growth pattern, with values doubling at each step. The consistent color gradient across both columns reinforces this trend. - **Anomalies**: No outliers are present; all values are evenly spaced and follow the expected progression. - **Design Choice**: The use of dot size to differentiate D_s and D_r (rather than a third axis) simplifies the visualization but may obscure quantitative differences in magnitude. --- ### Technical Notes - **Language**: English (no non-English text present). - **Data Structure**: - Two columns (D_s, D_r) with shared y-axis values. - Legend acts as a value-to-color mapping. - **Spatial Grounding**: - D_s: Left column, large dots. - D_r: Right column, small dots. - Legend: Right-aligned, vertical orientation. </details> | <details> <summary>x17.png Details</summary>  ### Visual Description ## Line Chart: Performance Speed-Up Analysis ### Overview The chart compares real-world and theoretical speed-ups in computational performance, measured by MFLOPS/Query (x-axis) and mAP@10% (y-axis). Two data series are represented: a green dashed line for "6x real-world speed-up" and an orange dashed line for "32x theoretical speed-up." Purple dots labeled "Funnel" are plotted along the green line, while blue dots appear below it. Red "Y" markers highlight specific points. ### Components/Axes - **X-axis**: "MFLOPS/Query" (log scale: 10² to 10⁴). - **Y-axis**: "mAP@10 (%)" (linear scale: 16.0 to 17.5). - **Legend**: Located in the bottom-right corner. - Red "Y": "Funnel" (data points). - Green dashed line: "6x real-world speed-up." - Orange dashed line: "32x theoretical speed-up." - **Annotations**: - "6x real-world speed-up" (green dashed line). - "32x theoretical speed-up" (orange dashed line). ### Detailed Analysis - **X-axis markers**: 10², 10³, 10⁴. - **Y-axis markers**: 16.0, 16.5, 17.0, 17.5. - **Data series**: - **Green dashed line**: Flat, indicating constant "6x real-world speed-up" across MFLOPS/Query values. - **Orange dashed line**: Starts at ~10³ MFLOPS/Query, rising steeply to ~10⁴, showing increasing "32x theoretical speed-up." - **Purple dots ("Funnel")**: Positioned along the green line at MFLOPS/Query values of ~10³ and ~10⁴, with mAP@10% values of ~17.0 and ~17.5. - **Blue dots**: Located below the green line, with MFLOPS/Query values ranging from ~10² to ~10³ and mAP@10% values between ~16.0 and ~16.5. - **Red "Y" markers**: Highlight specific points on the green line (e.g., ~10³ MFLOPS/Query, ~17.0 mAP@10%) and below it (e.g., ~10² MFLOPS/Query, ~16.0 mAP@10%). ### Key Observations 1. The green line ("6x real-world speed-up") remains flat, suggesting consistent performance across MFLOPS/Query. 2. The orange line ("32x theoretical speed-up") shows a sharp upward trend, indicating higher theoretical gains at higher MFLOPS/Query. 3. "Funnel" points (purple dots) align with the green line, implying these represent optimal real-world performance. 4. Blue dots (lower mAP@10%) may represent suboptimal or alternative configurations. 5. Red "Y" markers emphasize critical data points, possibly thresholds or benchmarks. ### Interpretation The chart highlights a disparity between real-world and theoretical performance. The "6x real-world speed-up" (green line) is stable, while the "32x theoretical speed-up" (orange line) suggests significant potential for improvement at higher computational loads. The "Funnel" points on the green line may indicate scenarios where real-world performance matches theoretical expectations, possibly due to optimized configurations. The blue dots below the green line could represent less efficient setups or edge cases. The red "Y" markers likely denote key evaluation points, such as baseline performance or target thresholds. This analysis underscores the gap between theoretical models and practical implementation, emphasizing the need for further optimization to bridge this gap. </details> | | --- | --- | --- | | (a) ImageNet-1K | | (b) ImageNet-4K | Figure 8: The trade-off between mAP@ $10$ vs MFLOPs/Query for Adaptive Retrieval (AR) on ImageNet-1K (left) and ImageNet-4K (right). Every combination of $D_{s}$ & $D_{r}$ falls above the Pareto line (orange dots) of single-shot retrieval with a fixed representation size while having configurations that are as accurate while being up to $14\times$ faster in real-world deployment. Funnel retrieval is almost as accurate as the baseline while alleviating some of the parameter choices of Adaptive Retrieval. Figure 8 showcases the compute-vs-accuracy trade-off for adaptive retrieval using ${\rm Matryoshka~Representations}$ compared to single-shot using fixed features with ResNet50 on ImageNet-1K. We observed that all AR settings lied above the Pareto frontier of single-shot retrieval with varying representation sizes. In particular for ImageNet-1K, we show that the AR model with $D_{s}=16$ & $D_{r}=2048$ is as accurate as single-shot retrieval with $d=2048$ while being $\mathbf{\sim 128\times}$ more efficient in theory and $\mathbf{\sim 14\times}$ faster in practice (compared using HNSW on the same hardware). We show similar trends with ImageNet-4K, but note that we require $D_{s}=64$ given the increased difficulty of the dataset. This results in $\sim 32\times$ and $\sim 6\times$ theoretical and in-practice speedups respectively. Lastly, while $K=200$ works well for our adaptive retrieval experiments, we ablated over the shortlist size $k$ in Appendix K.2 and found that the accuracy gains stopped after a point, further strengthening the use-case for ${\rm Matryoshka~Representation~Learning}$ and adaptive retrieval. Even with adaptive retrieval, it is hard to determine the choice of $D_{s}$ & $D_{r}$ . In order to alleviate this issue to an extent, we propose Funnel Retrieval, a consistent cascade for adaptive retrieval. Funnel thins out the initial shortlist by a repeated re-ranking and shortlisting with a series of increasing capacity representations. Funnel halves the shortlist size and doubles the representation size at every step of re-ranking. For example on ImageNet-1K, a funnel with the shortlist progression of $200\to 100\to 50\to 25\to 10$ with the cascade of $16\to 32\to 64\to 128\to 256\to 2048$ representation sizes within ${\rm Matryoshka~Representation}$ is as accurate as the single-shot 2048-dim retrieval while being $\sim 128\times$ more efficient theoretically (see Appendix F for more results). All these results showcase the potential of ${\rm MRL}$ and AR for large-scale multi-stage search systems [15]. ## 5 Further Analysis and Ablations Robustness. We evaluate the robustness of the ${\rm MRL}$ models trained on ImageNet-1K on out-of-domain datasets, ImageNetV2/R/A/Sketch [72, 34, 35, 94], and compare them to the FF baselines. Table 17 in Appendix H demonstrates that ${\rm Matryoshka~Representations}$ for classification are at least as robust as the original representation while improving the performance on ImageNet-A by $0.6\$ – a $20\$ relative improvement. We also study the robustness in the context of retrieval by using ImageNetV2 as the query set for ImageNet-1K database. Table 9 in Appendix E shows that ${\rm MRL}$ models have more robust retrieval compared to the FF baselines by having up to $3\$ higher mAP@ $10$ performance. This observation also suggests the need for further investigation into robustness using nearest neighbour based classification and retrieval instead of the standard linear probing setup. We also find that the zero-shot robustness of ALIGN- ${\rm MRL}$ (Table 18 in Appendix H) agrees with the observations made by Wortsman et al. [96]. Lastly, Table 6 in Appendix D.2 shows that ${\rm MRL}$ also improves the cosine similarity span between positive and random image-text pairs. Few-shot and Long-tail Learning. We exhaustively evaluated few-shot learning on ${\rm MRL}$ models using nearest class mean [79]. Table 15 in Appendix G shows that that representations learned through ${\rm MRL}$ perform comparably to FF representations across varying shots and number of classes. ${\rm Matryoshka~Representations}$ realize a unique pattern while evaluating on FLUID [92], a long-tail sequential learning framework. We observed that ${\rm MRL}$ provides up to $2\$ accuracy higher on novel classes in the tail of the distribution, without sacrificing accuracy on other classes (Table 16 in Appendix G). Additionally we find the accuracy between low-dimensional and high-dimensional representations is marginal for pretrain classes. We hypothesize that the higher-dimensional representations are required to differentiate the classes when few training examples of each are known. This results provides further evidence that different tasks require varying capacity based on their difficulty. | (a) (b) (c) | <details> <summary>TabsNFigs/images/gradcam-annotated-1.png Details</summary>  ### Visual Description ## Diagram: Attention Visualization for Object Recognition ### Overview The image depicts a sequence of attention heatmaps visualizing a model's focus progression across iterations. It includes a ground truth (GT) image of a person holding a plastic bag, followed by four heatmaps labeled with numbers (8, 16, 32, 2048) representing computational steps or iterations. Arrows indicate a conceptual flow from "Shower Cap" to "Plastic Bag," suggesting a correction in attention focus over time. ### Components/Axes - **Left Panel**: Ground truth (GT) image labeled "GT: Plastic Bag," showing a person holding a white plastic bag. - **Right Panels**: Four attention heatmaps with progressive numbers (8, 16, 32, 2048) in orange text at the bottom of each panel. - **Labels**: - "Shower Cap" (leftmost label, purple text) - "Plastic Bag" (rightmost label, black text) - **Arrows**: Two black arrows connecting "Shower Cap" → "Plastic Bag," indicating directional flow. - **Heatmap Colors**: Gradient from purple (low attention) to yellow (high attention), with no explicit legend. ### Detailed Analysis 1. **GT Image**: - Person wearing a brown jacket, red bag, and white hat, holding a white plastic bag. - Background includes pedestrians and urban elements (flowers, buildings). 2. **Heatmaps**: - **8**: Faint yellow glow around the plastic bag, indicating initial but weak focus. - **16**: Slightly stronger attention on the bag, with residual focus on the person's upper body. - **32**: Concentrated yellow highlight on the plastic bag, with reduced attention on the person. - **2048**: Dominant yellow focus on the bag, minimal attention elsewhere. 3. **Textual Elements**: - Numbers (8, 16, 32, 2048) are positioned at the bottom center of each heatmap in orange. - Labels "Shower Cap" and "Plastic Bag" are placed at the far left and right of the diagram, respectively. ### Key Observations - The heatmaps show a clear progression from diffuse attention (early iterations) to precise focus on the plastic bag (later iterations). - The "Shower Cap" label is spatially isolated from the heatmaps, suggesting it represents an initial misclassification or distracting element. - The numbers increase exponentially (8 → 2048), implying computational complexity or depth in the model's processing. ### Interpretation The diagram demonstrates how an attention mechanism refines object recognition over iterations. Initially, the model may misattribute focus to irrelevant elements (e.g., "Shower Cap"), but with increased computational steps, it prioritizes the ground truth object ("Plastic Bag"). The exponential growth in iteration numbers (8 → 2048) highlights the trade-off between precision and computational cost. The absence of a legend for heatmap colors suggests a standardized intensity scale (e.g., 0–1), with yellow representing maximum attention. This visualization underscores the importance of iterative refinement in attention-based models for accurate object localization. </details> <details> <summary>TabsNFigs/images/gradcam-annotated-2.png Details</summary>  ### Visual Description ## Heatmap Visualization: Snake Species Discrimination Progression ### Overview The image presents a comparative analysis of snake species discrimination using heatmap visualizations. It shows a progression from Boa Constrictor representations to a Rock Python ground truth (GT), with four intermediate stages labeled 8, 16, 32, and 2048. Each stage displays a heatmap overlay on a snake silhouette, with color gradients indicating activation intensity. ### Components/Axes - **Left Panel**: Ground Truth (GT) image of a Rock Python with natural patterning on a yellow background. - **Central Panels**: Four sequential heatmaps labeled 8, 16, 32, and 2048, showing: - **X-axis**: Implicit progression from Boa Constrictor (left) to Rock Python (right) via arrows. - **Y-axis**: Not explicitly labeled, but implied vertical dimension for heatmap distribution. - **Color Legend**: - Purple (low activation) - Green (moderate activation) - Yellow (high activation) - **Text Elements**: - "GT: Rock Python" (top-left) - "Boa Constrictor" (center-left) - "Rock Python" (center-right) - Numerical labels (8, 16, 32, 2048) in orange at bottom of each heatmap. ### Detailed Analysis 1. **Heatmap Progression**: - **8**: Broad activation across the snake's body, with diffuse green/yellow regions. - **16**: Increased focus on the head region, with concentrated yellow areas. - **32**: Further refinement, with distinct yellow patches along the dorsal ridge. - **2048**: Highly localized activation, with intense yellow concentrated on the head and ocular region. 2. **Color Distribution**: - All heatmaps show a gradient from purple (background) to yellow (peak activation). - Yellow regions correlate with anatomical features critical for species identification (head shape, scale patterns). 3. **Spatial Relationships**: - Arrows indicate a left-to-right progression from Boa Constrictor to Rock Python. - Heatmap intensity increases correlate with the numerical labels (8→2048). ### Key Observations - **Activation Concentration**: Higher numerical values (2048) show significantly more focused activation than lower values (8). - **Species Discrimination**: The rightmost heatmap (2048) most closely resembles the GT Rock Python's anatomical features. - **Color Consistency**: Yellow regions consistently align with the head/eye area across all stages. ### Interpretation This visualization demonstrates how increasing model complexity (represented by the numerical labels) improves feature discrimination between Boa Constrictors and Rock Pythons. The heatmaps likely represent: 1. **Attention Mechanisms**: Higher values show focused attention on diagnostic features (head shape, ocular region). 2. **Model Capacity**: The progression from 8→2048 suggests exponential scaling of model parameters/layers. 3. **Biological Relevance**: The final heatmap (2048) aligns with human-identifiable features of Rock Pythons, validating the model's discriminative capability. The data implies that model performance improves non-linearly with increased capacity, with the 2048-stage achieving near-ground-truth discrimination. This pattern is critical for optimizing computational resources in biological image classification tasks. </details> <details> <summary>TabsNFigs/images/gradcam-annotated-3.png Details</summary>  ### Visual Description ## Image Sequence: Visualization of Attention Maps ### Overview The image displays a sequence of five visualizations of a doll wearing a yellow sweatshirt, with progressive changes in visual effects. The sequence includes a ground truth (GT) image, followed by four heatmap-style representations labeled with numbers (8, 16, 32, 2048). Arrows indicate relationships between "GT: Sweatshirt," "Sunglasses," and "Sweatshirt," suggesting a progression or transformation process. ### Components/Axes - **Labels**: - "GT: Sweatshirt" (ground truth reference) - "Sunglasses" (left arrow) - "Sweatshirt" (right arrow) - **Numbers**: - 8, 16, 32, 2048 (exponentially increasing values, likely representing scale or intensity) - **Color Gradient**: - Blue (low intensity) → Green (medium) → Yellow (high intensity), indicating attention or activation levels. ### Detailed Analysis 1. **GT: Sweatshirt (Leftmost Image)**: - A doll with short blonde hair, large eyes, and a yellow sweatshirt against a blue background. - No visual effects; serves as a baseline reference. 2. **Sunglasses (Second Image, Number 8)**: - Overlay of green/yellow heatmap on the doll’s face and sweatshirt. - Brightest intensity on the sunglasses area (yellow), fading to green on the sweatshirt. 3. **Sweatshirt (Third Image, Number 16)**: - Similar heatmap but with broader coverage. - Sweatshirt area shows stronger yellow intensity, while sunglasses remain localized. 4. **Sweatshirt (Fourth Image, Number 32)**: - Heatmap expands further, with yellow dominating the sweatshirt and fading to green on the face. - Background transitions to purple, suggesting reduced attention to non-target regions. 5. **Sweatshirt (Fifth Image, Number 2048)**: - Most intense heatmap, with yellow concentrated on the sweatshirt and minimal green on the face. - Background remains purple, emphasizing focus on the target object. ### Key Observations - **Exponential Scaling**: Numbers (8, 16, 32, 2048) follow powers of 2, suggesting logarithmic scaling of attention or model parameters. - **Attention Localization**: - Lower values (8, 16) show mixed focus on sunglasses and sweatshirt. - Higher values (32, 2048) concentrate attention on the sweatshirt, aligning with the "Sweatshirt" label. - **Color Correlation**: - Yellow corresponds to highest intensity (target regions). - Green indicates moderate attention (secondary regions). - Blue/Purple represents low/no attention (background). ### Interpretation The sequence visualizes how an attention mechanism in a computational model (e.g., neural network) prioritizes specific regions (sunglasses and sweatshirt) as the scale increases. The ground truth (GT) establishes the target object (sweatshirt), while the heatmaps demonstrate the model’s ability to isolate and focus on this object. The exponential increase in numbers (8 → 2048) likely reflects iterative refinement or parameter adjustments, with higher values enabling sharper localization. The transition from mixed attention (sunglasses/sweatshirt) to singular focus (sweatshirt) suggests the model learns to prioritize the GT label over distractors. This could relate to tasks like object segmentation, where attention maps guide the model to distinguish foreground from background. </details> | | --- | --- | Figure 9: Grad-CAM [80] progression of predictions in ${\rm MRL}$ model across $8,16,32\text{ and }2048$ dimensions. (a) $8$ -dimensional representation confuses due to presence of other relevant objects (with a larger field of view) in the scene and predicts “shower cap” ; (b) $8$ -dim model confuses within the same super-class of “boa” ; (c) $8$ and $16$ -dim models incorrectly focus on the eyes of the doll ("sunglasses") and not the "sweatshirt" which is correctly in focus at higher dimensions; ${\rm MRL}$ fails gracefully in these scenarios and shows potential use cases of disagreement across dimensions. Disagreement across Dimensions. The information packing in ${\rm Matryoshka~Representations}$ often results in gradual increase of accuracy with increase in capacity. However, we observed that this trend was not ubiquitous and certain instances and classes were more accurate when evaluated with lower-dimensions (Figure 12 in Appendix J). With perfect routing of instances to appropriate dimension, ${\rm MRL}$ can gain up to $4.6\$ classification accuracy. At the same time, the low-dimensional models are less accurate either due to confusion within the same superclass [24] of the ImageNet hierarchy or presence of multiple objects of interest. Figure 9 showcases 2 such examples for $8$ -dimensional representation. These results along with Appendix J put forward the potential for ${\rm MRL}$ to be a systematic framework for analyzing the utility and efficiency of information bottlenecks. <details> <summary>x18.png Details</summary>  ### Visual Description ## Bar Chart: Top-1 Accuracy Comparison Between MRL and FF Across Representation Sizes ### Overview The chart compares the Top-1 Accuracy (%) of two methods, **MRL** (blue bars) and **FF** (orange bars), across varying **Representation Sizes** (8 to 2048). Both methods show increasing accuracy with larger representation sizes, with MRL consistently outperforming FF by small margins. ### Components/Axes - **X-axis**: Representation Size (logarithmic scale: 8, 16, 32, 64, 128, 256, 512, 1024, 2048). - **Y-axis**: Top-1 Accuracy (%) (range: 84% to 90%). - **Legend**: - Blue = MRL - Orange = FF - **Bar Groups**: Paired bars for each representation size, with MRL (blue) on the left and FF (orange) on the right. ### Detailed Analysis - **Representation Size 8**: - MRL: ~85.5% - FF: ~85.0% - **Representation Size 16**: - MRL: ~88.5% - FF: ~88.0% - **Representation Size 32**: - MRL: ~89.2% - FF: ~88.8% - **Representation Size 64**: - MRL: ~89.5% - FF: ~89.1% - **Representation Size 128**: - MRL: ~89.7% - FF: ~89.3% - **Representation Size 256**: - MRL: ~89.8% - FF: ~89.5% - **Representation Size 512**: - MRL: ~90.0% - FF: ~89.7% - **Representation Size 1024**: - MRL: ~90.2% - FF: ~90.0% - **Representation Size 2048**: - MRL: ~90.5% - FF: ~90.3% ### Key Observations 1. **Upward Trend**: Both methods show a clear increase in Top-1 Accuracy as representation size grows. 2. **Consistent MRL Advantage**: MRL outperforms FF by ~0.2–0.5% across all sizes, with the gap narrowing at larger scales (e.g., 0.2% at 2048). 3. **Diminishing Returns**: Accuracy improvements plateau at larger sizes (e.g., 1024–2048), suggesting limited gains beyond ~1024. ### Interpretation The data indicates that **MRL** is marginally more effective than **FF** for improving Top-1 Accuracy, particularly at smaller representation sizes. However, both methods converge in performance at larger scales (2048), implying that representation size has a diminishing impact on accuracy gains. This could reflect architectural optimizations in MRL or saturation of model capacity at higher dimensions. The trend underscores the importance of balancing representation size with computational efficiency in model design. </details> Figure 10: 31-way ImageNet-1K superclass classification across representation size for ${\rm MRL}$ & FF models showing the capture of underlying hierarchy through tight information bottlenecks. <details> <summary>x19.png Details</summary>  ### Visual Description ## Line Graph: Top-1 Accuracy (%) vs. Representation Size ### Overview The image is a line graph comparing the top-1 accuracy (%) of various categories across different representation sizes (8 to 2048). The graph includes eight data series, each represented by a distinct color and marker. The x-axis (Representation Size) uses logarithmic scaling (powers of 2), while the y-axis (Top-1 Accuracy %) ranges from 65% to 95%. ### Components/Axes - **X-axis (Representation Size)**: Logarithmic scale with values: 8, 16, 32, 64, 128, 256, 512, 1024, 2048. - **Y-axis (Top-1 Accuracy %)**: Linear scale from 65% to 95%. - **Legend**: Located in the bottom-right corner, mapping colors/markers to categories: - Blue solid line with circles: Measuring device - Red dashed line with triangles: Building - Green dashed line with triangles: Garment - Brown dotted line with crosses: Tool - Orange dashed line with diamonds: Nourishment - Purple solid line with stars: Protective covering - Pink dotted line with diamonds: Vessel - Teal solid line with stars: Oscine ### Detailed Analysis 1. **Measuring Device (Blue)**: Starts at ~94% (8), rises to ~95% (16), plateaus at ~96% (32–2048). 2. **Building (Red)**: Begins at ~90% (8), increases to ~95% (64), stabilizes at ~95% (128–2048). 3. **Garment (Green)**: Starts at ~65% (8), sharply rises to ~80% (64), then plateaus at ~85% (128–2048). 4. **Tool (Brown)**: Begins at ~70% (8), increases to ~85% (128), stabilizes at ~88% (256–2048). 5. **Nourishment (Orange)**: Starts at ~75% (8), rises to ~90% (2048), with steady growth across all sizes. 6. **Protective Covering (Purple)**: Begins at ~80% (8), increases to ~90% (128), plateaus at ~90% (256–2048). 7. **Vessel (Pink)**: Starts at ~85% (8), rises to ~90% (128), stabilizes at ~90% (256–2048). 8. **Oscine (Teal)**: Starts at ~95% (8), remains flat at ~95–96% (16–2048). ### Key Observations - **Highest Performers**: Oscine (teal) and measuring device (blue) maintain the highest accuracy (~95–96%) across all sizes. - **Most Improvement**: Garment (green) and tool (brown) show the steepest increases, starting at ~65% and ~70% (8) and reaching ~85% and ~88% (2048), respectively. - **Stable Lines**: Building (red), protective covering (purple), and vessel (pink) exhibit minimal changes after ~64–128 representation size. - **Gradual Growth**: Nourishment (orange) shows consistent improvement from ~75% to ~90% as representation size increases. ### Interpretation The data suggests that **larger representation sizes generally improve top-1 accuracy** for most categories, with **garment** and **tool** benefiting the most from increased size. **Oscine** and **measuring device** demonstrate robustness, maintaining high accuracy regardless of representation size. This implies that certain categories (e.g., garment, tool) may require higher-resolution representations to achieve optimal performance, while others (e.g., oscine) are less sensitive to size variations. The stability of lines like building and protective covering indicates diminishing returns at larger sizes for these categories. </details> Figure 11: Diverse per-superclass accuracy trends across representation sizes for ResNet50- ${\rm MRL}$ on ImageNet-1K. Superclass Accuracy. As the information bottleneck becomes smaller, the overall accuracy on fine-grained classes decreases rapidly (Figure 3). However, the drop-off is not as significant when evaluated at a superclass level (Table 24 in Appendix J). Figure 11 presents that this phenomenon occurs with both ${\rm MRL}$ and FF models; ${\rm MRL}$ is more accurate across dimensions. This shows that tight information bottlenecks while not highly accurate for fine-grained classification, do capture required semantic information for coarser classification that could be leveraged for adaptive routing for retrieval and classification. Mutifidelity of ${\rm Matryoshka~Representation}$ naturally captures the underlying hierarchy of the class labels with one single model. Lastly, Figure 11 showcases the accuracy trends per superclass with ${\rm MRL}$ . The utility of additional dimensions in distinguishing a class from others within the same superclass is evident for “garment” which has up to 11% improvement for 8 $\to$ 16 dimensional representation transition. We also observed that superclasses such as “oscine (songbird)” had a clear visual distinction between the object and background and thus predictions using 8 dimensions also led to a good inter-class separability within the superclass. ### 5.1 Ablations Table 26 in Appendix K presents that ${\rm Matryoshka~Representations}$ can be enabled within off-the-shelf pretrained models with inexpensive partial finetuning thus paving a way for ubiquitous adoption of ${\rm MRL}$ . At the same time, Table 27 in Appendix C indicates that with optimal weighting of the nested losses we could improve accuracy of lower-dimensions representations without accuracy loss. Tables 29 and 29 in Appendix C ablate over the choice of initial granularity and spacing of the granularites. Table 29 reaffirms the design choice to shun extremely low dimensions that have poor classification accuracy as initial granularity for ${\rm MRL}$ while Table 29 confirms the effectiveness of logarthmic granularity spacing inspired from the behaviour of accuracy saturation across dimensions over uniform. Lastly, Tables 30 and 31 in Appendix K.2 show that the retrieval performance saturates after a certain shortlist dimension and length depending on the complexity of the dataset. ## 6 Discussion and Conclusions The results in Section 5.1 reveal interesting weaknesses of ${\rm MRL}$ that would be logical directions for future work. (1) Optimizing the weightings of the nested losses to obtain a Pareto optimal accuracy-vs-efficiency trade-off – a potential solution could emerge from adaptive loss balancing aspects of anytime neural networks [41]. (2) Using different losses at various fidelities aimed at solving a specific aspect of adaptive deployment – e.g. high recall for $8$ -dimension and robustness for $2048$ -dimension. (3) Learning a search data-structure, like differentiable k-d tree, on top of ${\rm Matryoshka~Representation}$ to enable dataset and representation aware retrieval. (4) Finally, the joint optimization of multi-objective ${\rm MRL}$ combined with end-to-end learnable search data-structure to have data-driven adaptive large-scale retrieval for web-scale search applications. In conclusion, we presented <details> <summary>x20.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ), a flexible representation learning approach that encodes information at multiple granularities in a single embedding vector. This enables the ${\rm MRL}$ to adapt to a downstream task’s statistical complexity as well as the available compute resources. We demonstrate that ${\rm MRL}$ can be used for large-scale adaptive classification as well as adaptive retrieval. On standard benchmarks, ${\rm MRL}$ matches the accuracy of the fixed-feature baseline despite using $14\times$ smaller representation size on average. Furthermore, the ${\rm Matryoshka~Representation}$ based adaptive shortlisting and re-ranking system ensures comparable mAP@ $10$ to the baseline while being $128\times$ cheaper in FLOPs and $14\times$ faster in wall-clock time. Finally, most of the efficiency techniques for model inference and vector search are complementary to ${\rm MRL}$ <details> <summary>x21.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> further assisting in deployment at the compute-extreme environments. ## Acknowledgments We are grateful to Srinadh Bhojanapalli, Lovish Madaan, Raghav Somani, Ludwig Schmidt, and Venkata Sailesh Sanampudi for helpful discussions and feedback. Aditya Kusupati also thanks Tom Duerig and Rahul Sukthankar for their support. Part of the paper’s large-scale experimentation is supported through a research GCP credit award from Google Cloud and Google Research. Gantavya Bhatt is supported in part by the CONIX Research Center, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA. Sham Kakade acknowledges funding from the NSF award CCF-1703574 and ONR N00014-22-1-2377. Ali Farhadi acknowledges funding from the NSF awards IIS 1652052, IIS 17303166, DARPA N66001-19-2-4031, DARPA W911NF-15-1-0543 and gifts from Allen Institute for Artificial Intelligence. ## References - Abadi et al. [2015] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL https://www.tensorflow.org/. Software available from tensorflow.org. - Barbu et al. [2019] A. Barbu, D. Mayo, J. Alverio, W. Luo, C. Wang, D. Gutfreund, J. Tenenbaum, and B. Katz. Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models. Advances in neural information processing systems, 32, 2019. - Bengio et al. [2010] S. Bengio, J. Weston, and D. Grangier. Label embedding trees for large multi-class tasks. Advances in Neural Information Processing Systems, 23, 2010. - Bengio [2012] Y. Bengio. Deep learning of representations for unsupervised and transfer learning. In Proceedings of ICML workshop on unsupervised and transfer learning, pages 17–36. JMLR Workshop and Conference Proceedings, 2012. - Bentley [1990] J. L. Bentley. K-d trees for semidynamic point sets. In Proceedings of the sixth annual symposium on Computational geometry, pages 187–197, 1990. - Beygelzimer et al. [2006] A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In Proceedings of the 23rd international conference on Machine learning, pages 97–104, 2006. - Brin and Page [1998] S. Brin and L. Page. The anatomy of a large-scale hypertextual web search engine. Computer networks and ISDN systems, 30(1-7):107–117, 1998. - Brown et al. [2020] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020. - Cai et al. [2019] H. Cai, C. Gan, T. Wang, Z. Zhang, and S. Han. Once-for-all: Train one network and specialize it for efficient deployment. arXiv preprint arXiv:1908.09791, 2019. - Chang et al. [2020] W.-C. Chang, F. X. Yu, Y.-W. Chang, Y. Yang, and S. Kumar. Pre-training tasks for embedding-based large-scale retrieval. arXiv preprint arXiv:2002.03932, 2020. - Chang et al. [2021] W.-C. Chang, D. Jiang, H.-F. Yu, C. H. Teo, J. Zhang, K. Zhong, K. Kolluri, Q. Hu, N. Shandilya, V. Ievgrafov, et al. Extreme multi-label learning for semantic matching in product search. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 2643–2651, 2021. - Chen et al. [2020] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597–1607. PMLR, 2020. - Chen et al. [2021] Y. Chen, Z. Liu, H. Xu, T. Darrell, and X. Wang. Meta-baseline: exploring simple meta-learning for few-shot learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9062–9071, 2021. - Datar et al. [2004] M. Datar, N. Immorlica, P. Indyk, and V. S. Mirrokni. Locality-sensitive hashing scheme based on p-stable distributions. In Proceedings of the twentieth annual symposium on Computational geometry, pages 253–262, 2004. - Dean [2009] J. Dean. Challenges in building large-scale information retrieval systems. In Keynote of the 2nd ACM International Conference on Web Search and Data Mining (WSDM), volume 10, 2009. - Deng et al. [2009] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009. - Deng et al. [2011] J. Deng, A. C. Berg, and L. Fei-Fei. Hierarchical semantic indexing for large scale image retrieval. In CVPR 2011, pages 785–792. IEEE, 2011. - Desai and Johnson [2021] K. Desai and J. Johnson. Virtex: Learning visual representations from textual annotations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11162–11173, 2021. - Devlin et al. [2018] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018. - Dietterich and Bakiri [1994] T. G. Dietterich and G. Bakiri. Solving multiclass learning problems via error-correcting output codes. Journal of artificial intelligence research, 2:263–286, 1994. - Divvala et al. [2014] S. K. Divvala, A. Farhadi, and C. Guestrin. Learning everything about anything: Webly-supervised visual concept learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3270–3277, 2014. - Dosovitskiy et al. [2020] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. - Engelsma et al. [2022] J. J. Engelsma, A. K. Jain, and V. N. Boddeti. Hers: Homomorphically encrypted representation search. IEEE Transactions on Biometrics, Behavior, and Identity Science, 4(3):349–360, 2022. - Engstrom et al. [2019] L. Engstrom, A. Ilyas, H. Salman, S. Santurkar, and D. Tsipras. Robustness (python library), 2019. URL https://github.com/MadryLab/robustness. - Gholami et al. [2021] A. Gholami, S. Kim, Z. Dong, Z. Yao, M. W. Mahoney, and K. Keutzer. A survey of quantization methods for efficient neural network inference. arXiv preprint arXiv:2103.13630, 2021. - Gong et al. [2019] S. Gong, V. N. Boddeti, and A. K. Jain. On the intrinsic dimensionality of image representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3987–3996, 2019. - Gutmann and Hyvärinen [2010] M. Gutmann and A. Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 297–304. JMLR Workshop and Conference Proceedings, 2010. - Harris and Giachritsis [2000] M. G. Harris and C. D. Giachritsis. Coarse-grained information dominates fine-grained information in judgments of time-to-contact from retinal flow. Vision research, 40(6):601–611, 2000. - He et al. [2016] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016. - He et al. [2020] K. He, H. Fan, Y. Wu, S. Xie, and R. Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 9729–9738, 2020. - He et al. [2021] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick. Masked autoencoders are scalable vision learners. arXiv preprint arXiv:2111.06377, 2021. - Hegdé [2008] J. Hegdé. Time course of visual perception: coarse-to-fine processing and beyond. Progress in neurobiology, 84(4):405–439, 2008. - Hendrycks and Gimpel [2016] D. Hendrycks and K. Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. arXiv preprint arXiv:1610.02136, 2016. - Hendrycks et al. [2021a] D. Hendrycks, S. Basart, N. Mu, S. Kadavath, F. Wang, E. Dorundo, R. Desai, T. Zhu, S. Parajuli, M. Guo, et al. The many faces of robustness: A critical analysis of out-of-distribution generalization. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 8340–8349, 2021a. - Hendrycks et al. [2021b] D. Hendrycks, K. Zhao, S. Basart, J. Steinhardt, and D. Song. Natural adversarial examples. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15262–15271, 2021b. - Hooker et al. [2019] S. Hooker, A. Courville, G. Clark, Y. Dauphin, and A. Frome. What do compressed deep neural networks forget? arXiv preprint arXiv:1911.05248, 2019. - Hooker et al. [2020] S. Hooker, N. Moorosi, G. Clark, S. Bengio, and E. Denton. Characterising bias in compressed models. arXiv preprint arXiv:2010.03058, 2020. - Hotelling [1933] H. Hotelling. Analysis of a complex of statistical variables into principal components. Journal of educational psychology, 24(6):417, 1933. - Howard et al. [2017] A. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, M. Andreetto, and H. Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017. - Howard and Ruder [2018] J. Howard and S. Ruder. Universal language model fine-tuning for text classification. arXiv preprint arXiv:1801.06146, 2018. - Hu et al. [2019] H. Hu, D. Dey, M. Hebert, and J. A. Bagnell. Learning anytime predictions in neural networks via adaptive loss balancing. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3812–3821, 2019. - Indyk and Motwani [1998] P. Indyk and R. Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 604–613, 1998. - Jain et al. [2019] H. Jain, V. Balasubramanian, B. Chunduri, and M. Varma. Slice: Scalable linear extreme classifiers trained on 100 million labels for related searches. In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining, pages 528–536, 2019. - Jayaram Subramanya et al. [2019] S. Jayaram Subramanya, F. Devvrit, H. V. Simhadri, R. Krishnawamy, and R. Kadekodi. Diskann: Fast accurate billion-point nearest neighbor search on a single node. Advances in Neural Information Processing Systems, 32, 2019. - Jegou et al. [2010] H. Jegou, M. Douze, and C. Schmid. Product quantization for nearest neighbor search. IEEE transactions on pattern analysis and machine intelligence, 33(1):117–128, 2010. - Jia et al. [2021] C. Jia, Y. Yang, Y. Xia, Y.-T. Chen, Z. Parekh, H. Pham, Q. Le, Y.-H. Sung, Z. Li, and T. Duerig. Scaling up visual and vision-language representation learning with noisy text supervision. In International Conference on Machine Learning, pages 4904–4916. PMLR, 2021. - Johnson et al. [2019] J. Johnson, M. Douze, and H. Jégou. Billion-scale similarity search with GPUs. IEEE Transactions on Big Data, 7(3):535–547, 2019. - Johnson [1984] W. B. Johnson. Extensions of lipschitz mappings into a hilbert space. Contemp. Math., 26:189–206, 1984. - Jouppi et al. [2017] N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, A. Borchers, et al. In-datacenter performance analysis of a tensor processing unit. In Proceedings of the 44th annual international symposium on computer architecture, pages 1–12, 2017. - Kaz Sato [2021] T. C. Kaz Sato. Vertex ai matching engine. Microsoft AI Blog, 2021. URL https://cloud.google.com/blog/topics/developers-practitioners/find-anything-blazingly-fast-googles-vector-search-technology. - Krizhevsky et al. [2012] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems, 25, 2012. - Kulis et al. [2009] B. Kulis, P. Jain, and K. Grauman. Fast similarity search for learned metrics. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(12):2143–2157, 2009. - Kusupati et al. [2018] A. Kusupati, M. Singh, K. Bhatia, A. Kumar, P. Jain, and M. Varma. Fastgrnn: A fast, accurate, stable and tiny kilobyte sized gated recurrent neural network. Advances in Neural Information Processing Systems, 31, 2018. - Kusupati et al. [2020] A. Kusupati, V. Ramanujan, R. Somani, M. Wortsman, P. Jain, S. Kakade, and A. Farhadi. Soft threshold weight reparameterization for learnable sparsity. In International Conference on Machine Learning, pages 5544–5555. PMLR, 2020. - Kusupati et al. [2021] A. Kusupati, M. Wallingford, V. Ramanujan, R. Somani, J. S. Park, K. Pillutla, P. Jain, S. Kakade, and A. Farhadi. Llc: Accurate, multi-purpose learnt low-dimensional binary codes. Advances in Neural Information Processing Systems, 34, 2021. - Leclerc et al. [2022] G. Leclerc, A. Ilyas, L. Engstrom, S. M. Park, H. Salman, and A. Madry. ffcv. https://github.com/libffcv/ffcv/, 2022. commit 607d117. - LeCun et al. [2015] Y. LeCun, Y. Bengio, and G. Hinton. Deep learning. nature, 521(7553):436–444, 2015. - Lee et al. [2016] S. Lee, S. Purushwalkam Shiva Prakash, M. Cogswell, V. Ranjan, D. Crandall, and D. Batra. Stochastic multiple choice learning for training diverse deep ensembles. Advances in Neural Information Processing Systems, 29, 2016. - Li et al. [2018] C. Li, H. Farkhoor, R. Liu, and J. Yosinski. Measuring the intrinsic dimension of objective landscapes. arXiv preprint arXiv:1804.08838, 2018. - Linde et al. [1980] Y. Linde, A. Buzo, and R. Gray. An algorithm for vector quantizer design. IEEE Transactions on communications, 28(1):84–95, 1980. - Loshchilov and Hutter [2017] I. Loshchilov and F. Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017. - Malkov and Yashunin [2018] Y. A. Malkov and D. A. Yashunin. Efficient and robust approximate nearest neighbor search using hierarchical navigable small world graphs. IEEE transactions on pattern analysis and machine intelligence, 42(4):824–836, 2018. - Masci et al. [2011] J. Masci, U. Meier, D. Cireşan, and J. Schmidhuber. Stacked convolutional auto-encoders for hierarchical feature extraction. In International conference on artificial neural networks, pages 52–59. Springer, 2011. - Mitra et al. [2002] P. Mitra, C. Murthy, and S. K. Pal. Unsupervised feature selection using feature similarity. IEEE transactions on pattern analysis and machine intelligence, 24(3):301–312, 2002. - Nanda et al. [2023] V. Nanda, T. Speicher, J. P. Dickerson, S. Feizi, K. P. Gummadi, and A. Weller. Diffused redundancy in pre-trained representations. arXiv preprint arXiv:2306.00183, 2023. - Nayak [2019] P. Nayak. Understanding searches better than ever before. Google AI Blog, 2019. URL https://blog.google/products/search/search-language-understanding-bert/. - Paszke et al. [2019] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019. - Peters et al. [2018] M. E. Peters, M. Neumann, M. Iyyer, M. Gardner, C. Clark, K. Lee, and L. Zettlemoyer. Deep contextualized word representations. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pages 2227–2237, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-1202. URL https://aclanthology.org/N18-1202. - Prabhu et al. [2020] Y. Prabhu, A. Kusupati, N. Gupta, and M. Varma. Extreme regression for dynamic search advertising. In Proceedings of the 13th International Conference on Web Search and Data Mining, pages 456–464, 2020. - Radford et al. [2018] A. Radford, K. Narasimhan, T. Salimans, and I. Sutskever. Improving language understanding by generative pre-training. OpenAI Blog, 2018. URL https://openai.com/blog/language-unsupervised/. - Radford et al. [2021] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, pages 8748–8763. PMLR, 2021. - Recht et al. [2019] B. Recht, R. Roelofs, L. Schmidt, and V. Shankar. Do imagenet classifiers generalize to imagenet? In International Conference on Machine Learning, pages 5389–5400. PMLR, 2019. - Rippel et al. [2014] O. Rippel, M. Gelbart, and R. Adams. Learning ordered representations with nested dropout. In International Conference on Machine Learning, pages 1746–1754. PMLR, 2014. - Rissanen [1978] J. Rissanen. Modeling by shortest data description. Automatica, 14(5):465–471, 1978. - Ruder et al. [2019] S. Ruder, M. E. Peters, S. Swayamdipta, and T. Wolf. Transfer learning in natural language processing. In Proceedings of the 2019 conference of the North American chapter of the association for computational linguistics: Tutorials, pages 15–18, 2019. - Russakovsky et al. [2015] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015. - Salakhutdinov and Hinton [2007] R. Salakhutdinov and G. Hinton. Learning a nonlinear embedding by preserving class neighbourhood structure. In Artificial Intelligence and Statistics, pages 412–419. PMLR, 2007. - Salakhutdinov and Hinton [2009] R. Salakhutdinov and G. Hinton. Semantic hashing. International Journal of Approximate Reasoning, 50(7):969–978, 2009. - Sánchez et al. [1997] J. S. Sánchez, F. Pla, and F. J. Ferri. On the use of neighbourhood-based non-parametric classifiers. Pattern Recognition Letters, 18(11-13):1179–1186, 1997. - Selvaraju et al. [2017] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pages 618–626, 2017. - Shazeer and Stern [2018] N. Shazeer and M. Stern. Adafactor: Adaptive learning rates with sublinear memory cost. In International Conference on Machine Learning, pages 4596–4604. PMLR, 2018. - Simonyan and Zisserman [2014] K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. - Smith [2017] L. N. Smith. Cyclical learning rates for training neural networks. In 2017 IEEE winter conference on applications of computer vision (WACV), pages 464–472. IEEE, 2017. - Soudry et al. [2018] D. Soudry, E. Hoffer, M. S. Nacson, S. Gunasekar, and N. Srebro. The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1):2822–2878, 2018. - Sun et al. [2017] C. Sun, A. Shrivastava, S. Singh, and A. Gupta. Revisiting unreasonable effectiveness of data in deep learning era. In Proceedings of the IEEE international conference on computer vision, pages 843–852, 2017. - Sutskever et al. [2013] I. Sutskever, J. Martens, G. Dahl, and G. Hinton. On the importance of initialization and momentum in deep learning. In International conference on machine learning, pages 1139–1147. PMLR, 2013. - Tan and Le [2019] M. Tan and Q. Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International conference on machine learning, pages 6105–6114. PMLR, 2019. - Van Der Maaten et al. [2009] L. Van Der Maaten, E. Postma, J. Van den Herik, et al. Dimensionality reduction: a comparative. J Mach Learn Res, 10(66-71):13, 2009. - Varma [2019] M. Varma. Extreme classification. Communications of the ACM, 62(11):44–45, 2019. - Viola and Jones [2001] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In Proceedings of the 2001 IEEE computer society conference on computer vision and pattern recognition. CVPR 2001, volume 1, pages I–I. Ieee, 2001. - Waldburger [2019] C. Waldburger. As search needs evolve, microsoft makes ai tools for better search available to researchers and developers. Microsoft AI Blog, 2019. URL https://blogs.microsoft.com/ai/bing-vector-search/. - Wallingford et al. [2020] M. Wallingford, A. Kusupati, K. Alizadeh-Vahid, A. Walsman, A. Kembhavi, and A. Farhadi. Are we overfitting to experimental setups in recognition? arXiv preprint arXiv:2007.02519, 2020. - Wallingford et al. [2022] M. Wallingford, H. Li, A. Achille, A. Ravichandran, C. Fowlkes, R. Bhotika, and S. Soatto. Task adaptive parameter sharing for multi-task learning. arXiv preprint arXiv:2203.16708, 2022. - Wang et al. [2019] H. Wang, S. Ge, Z. Lipton, and E. P. Xing. Learning robust global representations by penalizing local predictive power. In Advances in Neural Information Processing Systems, pages 10506–10518, 2019. - Wang et al. [2020] X. Wang, D. Kondratyuk, K. M. Kitani, Y. Movshovitz-Attias, and E. Eban. Multiple networks are more efficient than one: Fast and accurate models via ensembles and cascades. arXiv preprint arXiv:2012.01988, 2020. - Wortsman et al. [2021] M. Wortsman, G. Ilharco, M. Li, J. W. Kim, H. Hajishirzi, A. Farhadi, H. Namkoong, and L. Schmidt. Robust fine-tuning of zero-shot models. arXiv preprint arXiv:2109.01903, 2021. - Wu et al. [2018] Z. Wu, Y. Xiong, S. Yu, and D. Lin. Unsupervised feature learning via non-parametric instance-level discrimination. arXiv preprint arXiv:1805.01978, 2018. - Yosinski et al. [2014] J. Yosinski, J. Clune, Y. Bengio, and H. Lipson. How transferable are features in deep neural networks? Advances in neural information processing systems, 27, 2014. - Yu et al. [2022] H.-F. Yu, K. Zhong, J. Zhang, W.-C. Chang, and I. S. Dhillon. Pecos: Prediction for enormous and correlated output spaces. Journal of Machine Learning Research, 23(98):1–32, 2022. - Yu et al. [2018] J. Yu, L. Yang, N. Xu, J. Yang, and T. Huang. Slimmable neural networks. arXiv preprint arXiv:1812.08928, 2018. - Zellers et al. [2022] R. Zellers, J. Lu, X. Lu, Y. Yu, Y. Zhao, M. Salehi, A. Kusupati, J. Hessel, A. Farhadi, and Y. Choi. Merlot reserve: Neural script knowledge through vision and language and sound. arXiv preprint arXiv:2201.02639, 2022. - Zhu et al. [2015] Y. Zhu, R. Kiros, R. Zemel, R. Salakhutdinov, R. Urtasun, A. Torralba, and S. Fidler. Aligning books and movies: Towards story-like visual explanations by watching movies and reading books. In Proceedings of the IEEE international conference on computer vision, pages 19–27, 2015. ## Checklist 1. For all authors… 1. Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] 1. Did you describe the limitations of your work? [Yes] See Section 6 1. Did you discuss any potential negative societal impacts of your work? [N/A] Our work does not have any additional negative societal impact on top of the existing impact of representation learning. However, a study on the trade-off between representation size and the tendency to encode biases is an interesting future direction along the lines of existing literature [36, 37]. A part of this is already presented in Section 5. 1. Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 1. If you are including theoretical results… 1. Did you state the full set of assumptions of all theoretical results? [N/A] 1. Did you include complete proofs of all theoretical results? [N/A] 1. If you ran experiments… 1. Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See supplemental material and Appendix A. All the code and public models will be open sourced. 1. Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4 and Appendix C. 1. Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We benchmarked on large-scale datasets like ImageNet-1K, JFT-300M and ALIGN data with models like ResNet and ViT making it extremely expensive to run things multiple times. 1. Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C and Appendix I. 1. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets… 1. If your work uses existing assets, did you cite the creators? [Yes] 1. Did you mention the license of the assets? [No] All the non-proprietary datasets and code used are public under MIT, BSD or CC licenses. 1. Did you include any new assets either in the supplemental material or as a URL? [Yes] We created a new subset of ImageNet-21K for downstream evaluation of retrieval performance at scale. See Section 4.3 and Appendix B 1. Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] 1. Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] 1. If you used crowdsourcing or conducted research with human subjects… 1. Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] 1. Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] 1. Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Contents 1. 1 Introduction 1. 2 Related Work 1. 3 <details> <summary>x22.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ${\rm Matryoshka~Representation~Learning}$ 1. 4 Applications 1. 4.1 Representation Learning 1. 4.2 Classification 1. 4.2.1 Adaptive Classification 1. 4.3 Retrieval 1. 4.3.1 Adaptive Retrieval 1. 5 Further Analysis and Ablations 1. 5.1 Ablations 1. 6 Discussion and Conclusions 1. A Code for ${\rm Matryoshka~Representation~Learning}$ <details> <summary>x23.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ( ${\rm MRL}$ ) 1. B Datasets 1. C ${\rm Matryoshka~Representation~Learning}$ Model Training 1. D Classification Results 1. D.1 Adaptive Classification ( ${\rm MRL}$ –AC) 1. D.2 JFT, ALIGN and BERT 1. E Image Retrieval 1. F Adaptive Retrieval 1. G Few-shot and Sample Efficiency 1. H Robustness Experiments 1. I In Practice Costs 1. J Analysis of Model Disagreement 1. K Ablation Studies 1. K.1 ${\rm MRL}$ Training Paradigm 1. K.2 Retrieval ## Appendix A Code for ${\rm Matryoshka~Representation~Learning}$ <details> <summary>x24.png Details</summary>  ### Visual Description Icon/Small Image (28x28) </details> ( ${\rm MRL}$ ) We use Alg 1 and 2 provided below to train supervised ResNet50– ${\rm MRL}$ models on ImageNet-1K. We provide this code as a template to extend ${\rm MRL}$ to any domain. Algorithm 1 Pytorch code for ${\rm Matryoshka}$ Cross-Entropy Loss ⬇ class Matryoshka_CE_Loss (nn. Module): def __init__ (self, relative_importance, ** kwargs): super (Matryoshka_CE_Loss, self). __init__ () self. criterion = nn. CrossEntropyLoss (** kwargs) self. relative_importance = relative_importance # usually set to all ones def forward (self, output, target): loss =0 for i in range (len (output)): loss += self. relative_importance [i] * self. criterion (output [i], target) return loss Algorithm 2 Pytorch code for ${\rm MRL}$ Linear Layer ⬇ class MRL_Linear_Layer (nn. Module): def __init__ (self, nesting_list: List, num_classes =1000, efficient = False, ** kwargs): super (MRL_Linear_Layer, self). __init__ () self. nesting_list = nesting_list # set of m in M (Eq. 1) self. num_classes = num_classes self. is_efficient = efficient # flag for MRL-E if not is_efficient: for i, num_feat in enumerate (self. nesting_list): setattr (self, f "nesting_classifier_{i}", nn. Linear (num_feat, self. num_classes, ** kwargs)) else: setattr (self, "nesting_classifier_0", nn. Linear (self. nesting_list [-1], self. num_classes, ** kwargs)) # Instantiating one nn.Linear layer for MRL-E def forward (self, x): nesting_logits = () for i, num_feat in enumerate (self. nesting_list): if (self. is_efficient): efficient_logit = torch. matmul (x [:, : num_feat], (self. nesting_classifier_0. weight [:, : num_feat]). t ()) else: nesting_logits. append (getattr (self, f "nesting_classifier_{i}")(x [:, : num_feat])) if (self. is_efficient): nesting_logits. append (efficient_logit) return nesting_logits ## Appendix B Datasets ImageNet-1K [76] contains 1,281,167 labeled train images, and 50,000 labelled validation images across 1,000 classes. The images were transformed with standard procedures detailed by FFCV [56]. ImageNet-4K dataset was constructed by selecting 4,202 classes, non-overlapping with ImageNet-1K, from ImageNet-21K [16] with 1,050 or more examples. The train set contains 1,000 examples and the query/validation set contains 50 examples per class totalling to $\sim$ 4.2M and $\sim$ 200K respectively. We will release the list of images curated together to construct ImageNet-4K. JFT-300M [85] is a large-scale multi-label dataset with 300M images labelled across 18,291 categories. ALIGN [46] utilizes a large scale noisy image-text dataset containing 1.8B image-text pairs. ImageNet Robustness Datasets We experimented on the following datasets to examine the robustness of ${\rm MRL}$ models: ImageNetV2 [72] is a collection of 10K images sampled a decade after the original construction of ImageNet [16]. ImageNetV2 contains 10 examples each from the 1,000 classes of ImageNet-1K. ImageNet-A [35] contains 7.5K real-world adversarially filtered images from 200 ImageNet-1K classes. ImageNet-R [34] contains 30K artistic image renditions for 200 of the original ImageNet-1K classes. ImageNet-Sketch [94] contains 50K sketches, evenly distributed over all 1,000 ImageNet-1K classes. ObjectNet [2] contains 50K images across 313 object classes, each containing $\sim$ 160 images each. ## Appendix C ${\rm Matryoshka~Representation~Learning}$ Model Training We trained all ResNet50– ${\rm MRL}$ models using the efficient dataloaders of FFCV [56]. We utilized the rn50_40_epochs.yaml configuration file of FFCV to train all ${\rm MRL}$ models defined below: - ${\rm MRL}$ : ResNet50 model with the fc layer replaced by MRL_Linear_Layer (efficient = False) - ${\rm MRL\text{--}E}$ : ResNet50 model with the fc layer replaced by MRL_Linear_Layer (efficient = True) - FF–k: ResNet50 model with the fc layer replaced by torch. nn. Linear (k, num_classes), where k $\in[8,16,32,64,128,256,512,1024,2048]$ . We will henceforth refer to these models as simply FF, with the k value denoting representation size. We trained all ResNet50 models with a learning rate of $0.475$ with a cyclic learning rate schedule [83]. This was after appropriate scaling (0.25 $\times$ ) of the learning rate specified in the configuration file to accommodate for 2xA100 NVIDIA GPUs available for training, compared to the 8xA100 GPUs utilized in the FFCV benchmarks. We trained with a batch size of 256 per GPU, momentum [86] of 0.9, and an SGD optimizer with a weight decay of 1e-4. Our code (Appendix A) makes minimal modifications to the training pipeline provided by FFCV to learn ${\rm Matryoshka~Representations}$ . We trained ViT-B/16 models for JFT-300M on a 8x8 cloud TPU pod [49] using Tensorflow [1] with a batchsize of 128 and trained for 300K steps. Similarly, ALIGN models were trained using Tensorflow on 8x8 cloud TPU pod for 1M steps with a batchsize of 64 per TPU. Both these models were trained with adafactor optimizer [81] with a linear learning rate decay starting at 1e-3. Lastly, we trained a BERT-Base model on English Wikipedia and BookCorpus. We trained our models in Tensorflow using a 4x4 cloud TPU pod with a total batchsize of 1024. We used AdamW [61] optimizer with a linear learning rate decay starting at 1e-4 and trained for 450K steps. In each configuration/case, if the final representation was normalized in the FF implementation, ${\rm MRL}$ models adopted the same for each nested dimension for a fair comparison. ## Appendix D Classification Results Table 1: Top-1 classification accuracy (%) for ResNet50 ${\rm MRL}$ and baseline models on ImageNet-1K. | 8 16 32 | 4.56 11.29 27.21 | 2.34 7.17 20.46 | 65.29 72.85 74.60 | 0.42 0.96 2.27 | 66.63 73.53 75.03 | 56.66 71.94 74.48 | | --- | --- | --- | --- | --- | --- | --- | | 64 | 49.47 | 48.10 | 75.27 | 5.59 | 75.82 | 75.35 | | 128 | 65.70 | 67.24 | 75.29 | 14.15 | 76.30 | 75.80 | | 256 | 72.43 | 74.59 | 75.71 | 38.42 | 76.47 | 76.22 | | 512 | 74.94 | 76.78 | 76.18 | 69.80 | 76.65 | 76.36 | | 1024 | 76.10 | 76.87 | 76.63 | 74.61 | 76.76 | 76.48 | | 2048 | 76.87 | – | 76.87 | 76.26 | 76.80 | 76.51 | We show the top-1 classification accuracy of ResNet50– ${\rm MRL}$ models on ImageNet-1K in Table 1 and Figure 3. We compare the performance of ${\rm MRL}$ models ( ${\rm MRL}$ , ${\rm MRL\text{--}E}$ ) to several baselines: - FF: We utilize the FF-k models described in Appendix C for $k\in\{8,...2048\}$ . - SVD: We performed a low rank approximation of the 1000-way classification layer of FF-2048, with rank = 1000. - Rand. LP: We compared against a linear classifier fit on randomly selected features [30]. - Slim. Net: We take pretrained slimmable neural networks [100] which are trained with a flexible width backbone (25%, 50%, 75% and full width). For each representation size, we consider the first $k$ dimensions for classification. Note that training of slimmable neural networks becomes unstable when trained below 25% width due to the hardness in optimization and low complexity of the model. At lower dimensions ( $d\leq 128$ ), ${\rm MRL}$ outperforms all baselines significantly, which indicates that pretrained models lack the multifidelity of ${\rm Matryoshka~Representations}$ and are incapable of fitting an accurate linear classifier at low representation sizes. We compared the performance of ${\rm MRL}$ models at various representation sizes via 1-nearest neighbors (1-NN) image classification accuracy on ImageNet-1K in Table 2 and Figure 3. We provide detailed information regarding the k-NN search pipeline in Appendix E. We compared against a baseline of attempting to enforce nesting to a FF-2048 model by 1) Random Feature Selection (Rand. FS): considering the first $m$ dimensions of FF-2048 for NN lookup, and 2) FF+SVD: performing SVD on the FF-2048 representations at the specified representation size, 3) FF+JL: performing random projection according to the Johnson-Lindenstrauss lemma [48] on the FF-2048 representations at the specified representation size. We also compared against the 1-NN accuracy of slimmable neural nets [100] as an additional baseline. We observed these baseline models to perform very poorly at lower dimensions, as they were not explicitly trained to learn ${\rm Matryoshka~Representations}$ . Table 2: 1-NN accuracy (%) on ImageNet-1K for various ResNet50 models. | 8 16 32 | 2.36 12.06 32.91 | 19.14 46.02 60.78 | 0.11 0.09 0.06 | 58.93 66.77 68.84 | 1.00 5.12 16.95 | 62.19 67.91 69.46 | 57.45 67.05 68.6 | | --- | --- | --- | --- | --- | --- | --- | --- | | 64 | 49.91 | 67.04 | 0.05 | 69.41 | 35.60 | 70.17 | 69.61 | | 128 | 60.91 | 69.63 | 0.06 | 69.35 | 51.16 | 70.52 | 70.12 | | 256 | 65.75 | 70.67 | 0.04 | 69.72 | 60.61 | 70.62 | 70.36 | | 512 | 68.77 | 71.06 | 0.03 | 70.18 | 65.82 | 70.82 | 70.74 | | 1024 | 70.41 | 71.22 | - | 70.34 | 67.19 | 70.89 | 71.07 | | 2048 | 71.19 | 71.21 | - | 71.19 | 66.10 | 70.97 | 71.21 | ### D.1 Adaptive Classification ( ${\rm MRL}$ –AC) Table 3: Threshold-based adaptive classification performance of ResNet50 ${\rm MRL}$ on a 40K sized held-out subset of the ImageNet-1K validation set. Results are averaged over 30 random held-out subsets. | 13.43 $\pm$ 0.81 | 73.79 $\pm$ 0.10 | | --- | --- | | 18.32 $\pm$ 1.36 | 75.25 $\pm$ 0.11 | | 25.87 $\pm$ 2.41 | 76.05 $\pm$ 0.15 | | 36.26 $\pm$ 4.78 | 76.28 $\pm$ 0.16 | | 48.00 $\pm$ 8.24 | 76.43 $\pm$ 0.18 | | 64.39 $\pm$ 12.55 | 76.53 $\pm$ 0.19 | | 90.22 $\pm$ 20.88 | 76.55 $\pm$ 0.20 | | 118.85 $\pm$ 33.37 | 76.56 $\pm$ 0.20 | In an attempt to use the smallest representation that works well for classification for every image in the ImageNet-1K validation set, we learned a policy to increase the representation size from $m_{i}$ to $m_{i+1}$ using a 10K sized subset of the ImageNet-1K validation set. This policy is based on whether the prediction confidence $p_{i}$ using representation size $m_{i}$ exceeds a learned threshold $t_{i}^{\ast}$ . If $p_{i}\geq t_{i}^{\ast}$ , we used predictions from representation size $m_{i}$ otherwise, we increased to representation size $m_{i+1}$ . To learn the optimal threshold $t_{i}^{\ast}$ , we performed a grid search between 0 and 1 (100 samples). For each threshold $t_{k}$ , we computed the classification accuracy over our 10K image subset. We set $t_{i}^{\ast}$ equal to the smallest threshold $t_{k}$ that gave the best accuracy. We use this procedure to obtain thresholds for successive models, i.e., $\{t_{j}^{\ast}\mid j\in\{8,16,32,64,\ldots,2048\}\}$ . To improve reliability of threshold based greedy policy, we use test time augmentation which has been used successfully in the past [82]. For inference, we used the remaining held-out 40K samples from the ImageNet-1K validation set. We began with smallest sized representation ( $m=8$ ) and compared the computed prediction confidence $p_{8}$ to learned optimal threshold $t_{8}^{\ast}$ . If $p_{8}\leq t_{8}^{\ast}$ , then we increased $m=16$ , and repeated this procedure until $m=d=2048$ . To compute the expected dimensions, we performed early stopping at $m=\{16,32,64,\ldots 2048\}$ and computed the expectation using the distribution of representation sizes. As shown in Table 3 and Figure 7, we observed that in expectation, we only needed a $\sim 37$ sized representation to achieve $76.3\$ classification accuracy on ImageNet-1K, which was roughly $14\times$ smaller than the FF–512 baseline. Even if we computed the expectation as a weighted average over the cumulative sum of representation sizes $\{8,24,56,\ldots\}$ , due to the nature of multiple linear heads for ${\rm MRL}$ , we ended up with an expected size of $62$ that still provided a roughly $8.2\times$ efficient representation than the FF–512 baseline. However, ${\rm MRL\text{--}E}$ alleviates this extra compute with a minimal drop in accuracy. ### D.2 JFT, ALIGN and BERT We examine the k-NN classification accuracy of learned ${\rm Matryoshka~Representations}$ via ALIGN– ${\rm MRL}$ and JFT-ViT– ${\rm MRL}$ in Table 4. For ALIGN [46], we observed that learning ${\rm Matryoshka~Representations}$ via ALIGN– ${\rm MRL}$ improved classification accuracy at nearly all dimensions when compared to ALIGN. We observed a similar trend when training ViT-B/16 [22] for JFT-300M [85] classification, where learning ${\rm Matryoshka~Representations}$ via ${\rm MRL}$ and ${\rm MRL\text{--}E}$ on top of JFT-ViT improved classification accuracy for nearly all dimensions, and significantly for lower ones. This demonstrates that training to learn ${\rm Matryoshka~Representations}$ is feasible and extendable even for extremely large scale datasets. We also demonstrate that ${\rm Matryoshka~Representations}$ are learned at interpolated dimensions for both ALIGN and JFT-ViT, as shown in Table 5, despite not being trained explicitly at these dimensions. Lastly, Table 6 shows that ${\rm MRL}$ training leads to a increase in the cosine similarity span between positive and random image-text pairs. Table 4: ViT-B/16 and ViT-B/16- ${\rm MRL}$ top-1 and top-5 k-NN accuracy (%) for ALIGN and JFT. Top-1 entries where ${\rm MRL\text{--}E}$ and ${\rm MRL}$ outperform baselines are bolded for both ALIGN and JFT-ViT. | 12 | 11.90 | 28.05 | 43.57 | 67.36 | 27.07 | 48.57 | 53.61 | 75.30 | 51.54 | 73.94 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 24 | 33.35 | 55.58 | 56.44 | 78.19 | 48.64 | 70.20 | 62.80 | 81.51 | 62.40 | 81.36 | | 48 | 51.32 | 73.15 | 62.33 | 82.30 | 63.58 | 81.80 | 67.24 | 84.37 | 66.89 | 83.80 | | 96 | 61.82 | 81.97 | 65.72 | 84.61 | 68.56 | 85.13 | 69.74 | 85.86 | 68.80 | 85.13 | | 192 | 66.71 | 85.27 | 67.00 | 85.36 | 71.32 | 86.21 | 71.34 | 86.62 | 70.41 | 86.01 | | 384 | 67.65 | 85.70 | 67.70 | 85.73 | 71.67 | 86.98 | 71.73 | 87.08 | 71.18 | 86.46 | | 768 | 68.00 | 86.10 | 67.85 | 85.85 | 72.10 | 87.20 | 71.85 | 86.92 | 71.31 | 86.62 | Table 5: Examining top-1 and top-5 k-NN accuracy (%) at interpolated hidden dimensions for ALIGN and JFT. This indicates that ${\rm MRL}$ is able to scale classification accuracy as hidden dimensions increase even at dimensions that were not explicitly considered during training. | 16 32 64 | 49.06 58.64 63.90 | 72.26 79.96 83.39 | 58.35 64.98 68.19 | 78.55 82.89 84.85 | | --- | --- | --- | --- | --- | | 128 | 66.63 | 85.00 | 70.35 | 86.24 | | 256 | 67.10 | 85.30 | 71.57 | 86.77 | | 512 | 67.64 | 85.72 | 71.55 | 86.67 | Table 6: Cosine similarity between embeddings | Positive Text to Image Random Text to Image Random Image to Image | 0.27 8e-3 0.10 | 0.49 -4e-03 0.08 | | --- | --- | --- | | Random Text to Text | 0.22 | 0.07 | We also evaluated the capability of ${\rm Matryoshka~Representations}$ to extend to other natural language processing via masked language modeling (MLM) with BERT [19], whose results are tabulated in Table 7. Without any hyper-parameter tuning, we observed ${\rm Matryoshka~Representations}$ to be within $0.5\$ of FF representations for BERT MLM validation accuracy. This is a promising initial result that could help with large-scale adaptive document retrieval using BERT– ${\rm MRL}$ . Table 7: Masked Language Modelling (MLM) accuracy(%) of FF and ${\rm MRL}$ models on the validation set. | 12 24 48 | 60.12 62.49 63.85 | 59.92 62.05 63.40 | | --- | --- | --- | | 96 | 64.32 | 64.15 | | 192 | 64.70 | 64.58 | | 384 | 65.03 | 64.81 | | 768 | 65.54 | 65.00 | ## Appendix E Image Retrieval We evaluated the strength of ${\rm Matryoshka~Representations}$ via image retrieval on ImageNet-1K (the training distribution), as well as on out-of-domain datasets ImageNetV2 and ImageNet-4K for all ${\rm MRL}$ ResNet50 models. We generated the database and query sets, containing $N$ and $Q$ samples respectively, with a standard PyTorch [67] forward pass on each dataset. We specify the representation size at which we retrieve a shortlist of k-nearest neighbors (k-NN) by $D_{s}$ . The database is a thus a [ $N$ , $D_{s}$ ] array, the query set is a [ $Q$ , $D_{s}$ ] array, and the neighbors set is a [ $Q$ , k] array. For metrics, we utilized corrected mean average precision (mAP@k) [55] and precision (P@k): $P@k=\dfrac{correct\_pred}{k}$ where $correct\_pred$ is the average number of retrieved NN with the correct label over the entire query set using a shortlist of length $k$ . We performed retrieval with FAISS [47], a library for efficient similarity search. To obtain a shortlist of k-NN, we built an index to search the database. We performed an exhaustive NN search with the L2 distance metric with faiss. IndexFlatL2, as well as an approximate NN search (ANNS) via HNSW [47] with faiss. IndexHNSWFlat. We used HNSW with $M=32$ unless otherwise mentioned, and henceforth referred to as HNSW32. The exact search index was moved to the GPU for fast k-NN search computation, whereas the HNSW index was kept on the CPU as it currently lacks GPU support. We show the wall clock times for building the index as well as the index size in Table 20. We observed exact search to have a smaller index size which was faster to build when compared to HNSW, which trades off a larger index footprint for fast NN search (discussed in more detail in Appendix K). The database and query vectors are normalized with faiss. normalize_L2 before building the index and performing search. Table 8: Retrieve a shortlist of 200-NN with $D_{s}$ sized representations on ImageNet-1K via exact search with L2 distance metric. Top-1 and mAP@10 entries (%) where ${\rm MRL\text{--}E}$ and ${\rm MRL}$ outperform FF at their respective representation sizes are bolded. | FF | 8 | 10 | 58.93 | 75.76 | 80.25 | 53.42 | 52.29 | 51.84 | 51.57 | 59.32 | 59.28 | 59.25 | 59.21 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 16 | 20 | 66.77 | 80.88 | 84.40 | 61.63 | 60.51 | 59.98 | 59.62 | 66.76 | 66.58 | 66.43 | 66.27 | | | 32 | 41 | 68.84 | 82.58 | 86.14 | 63.35 | 62.08 | 61.36 | 60.76 | 68.43 | 68.13 | 67.83 | 67.48 | | | 64 | 82 | 69.41 | 83.56 | 87.33 | 63.26 | 61.64 | 60.63 | 59.67 | 68.49 | 67.91 | 67.38 | 66.74 | | | 128 | 164 | 69.35 | 84.23 | 88.24 | 62.30 | 60.16 | 58.73 | 57.29 | 67.84 | 66.83 | 65.96 | 64.92 | | | 256 | 328 | 69.72 | 84.71 | 88.54 | 61.47 | 58.85 | 57.02 | 55.13 | 67.19 | 65.82 | 64.64 | 63.24 | | | 512 | 656 | 70.18 | 85.04 | 88.91 | 61.37 | 58.41 | 56.26 | 53.98 | 67.12 | 65.49 | 64.07 | 62.35 | | | 1024 | 1312 | 70.34 | 85.38 | 89.19 | 61.13 | 57.87 | 55.47 | 52.90 | 66.93 | 65.08 | 63.43 | 61.45 | | | 2048 | 2624 | 71.19 | 85.66 | 89.17 | 62.90 | 60.06 | 57.99 | 55.76 | 68.46 | 66.9 | 65.52 | 63.83 | | | ${\rm MRL\text{--}E}$ | 8 | 10 | 57.39 | 74.18 | 79.16 | 51.80 | 50.41 | 49.60 | 48.86 | 57.50 | 57.16 | 56.81 | 56.36 | | 16 | 20 | 67.08 | 81.38 | 85.15 | 61.60 | 60.36 | 59.66 | 59.04 | 66.79 | 66.53 | 66.24 | 65.87 | | | 32 | 41 | 68.62 | 82.92 | 86.44 | 63.34 | 61.97 | 61.14 | 60.39 | 68.49 | 68.06 | 67.65 | 67.17 | | | 64 | 82 | 69.56 | 83.49 | 86.85 | 63.84 | 62.33 | 61.43 | 60.57 | 68.93 | 68.4 | 67.96 | 67.38 | | | 128 | 164 | 70.13 | 83.63 | 87.07 | 64.15 | 62.58 | 61.61 | 60.70 | 69.19 | 68.62 | 68.11 | 67.50 | | | 256 | 328 | 70.39 | 83.8 | 87.28 | 64.35 | 62.76 | 61.76 | 60.82 | 69.36 | 68.79 | 68.26 | 67.63 | | | 512 | 656 | 70.74 | 83.91 | 87.33 | 64.69 | 63.05 | 62.06 | 61.14 | 69.63 | 69.00 | 68.50 | 67.88 | | | 1024 | 1312 | 71.05 | 84.13 | 87.46 | 64.85 | 63.22 | 62.19 | 61.26 | 69.78 | 69.16 | 68.60 | 67.99 | | | 2048 | 2624 | 71.17 | 84.27 | 87.67 | 64.99 | 63.33 | 62.29 | 61.33 | 69.90 | 69.24 | 68.68 | 68.05 | | | ${\rm MRL\text{--}E}$ Interpolated | 12 | 15 | 64.25 | 79.21 | 83.29 | 58.83 | 57.50 | 56.71 | 56.02 | 64.10 | 63.78 | 63.42 | 63.02 | | 24 | 31 | 68.28 | 82.31 | 85.89 | 62.75 | 61.41 | 60.62 | 59.92 | 67.89 | 67.49 | 67.11 | 66.69 | | | 48 | 61 | 69.20 | 83.15 | 86.67 | 63.58 | 62.12 | 61.23 | 60.42 | 68.71 | 68.19 | 67.75 | 67.22 | | | 96 | 123 | 70.05 | 83.63 | 87.11 | 64.04 | 62.46 | 61.52 | 60.63 | 69.10 | 68.51 | 68.04 | 67.45 | | | 192 | 246 | 70.36 | 83.72 | 87.21 | 64.26 | 62.65 | 61.65 | 60.72 | 69.26 | 68.67 | 68.15 | 67.53 | | | 384 | 492 | 70.54 | 83.88 | 87.28 | 64.55 | 62.94 | 61.93 | 61.01 | 69.51 | 68.92 | 68.40 | 67.78 | | | 768 | 984 | 70.96 | 84.05 | 87.44 | 64.79 | 63.15 | 62.15 | 61.22 | 69.72 | 69.10 | 68.56 | 67.95 | | | 1536 | 1968 | 71.19 | 84.17 | 87.57 | 64.94 | 63.29 | 62.26 | 61.32 | 69.85 | 69.21 | 68.66 | 68.04 | | | ${\rm MRL}$ | 8 | 10 | 62.19 | 77.05 | 81.34 | 56.74 | 55.47 | 54.76 | 54.12 | 62.06 | 61.81 | 61.54 | 61.17 | | 16 | 20 | 67.91 | 81.44 | 85.00 | 62.94 | 61.79 | 61.16 | 60.64 | 67.93 | 67.71 | 67.48 | 67.20 | | | 32 | 41 | 69.46 | 83.01 | 86.30 | 64.21 | 62.96 | 62.22 | 61.58 | 69.18 | 68.87 | 68.54 | 68.17 | | | 64 | 82 | 70.17 | 83.53 | 86.95 | 64.69 | 63.33 | 62.53 | 61.80 | 69.67 | 69.25 | 68.89 | 68.42 | | | 128 | 164 | 70.52 | 83.98 | 87.25 | 64.94 | 63.50 | 62.63 | 61.83 | 69.93 | 69.44 | 69.02 | 68.50 | | | 256 | 328 | 70.62 | 84.17 | 87.38 | 65.04 | 63.56 | 62.66 | 61.81 | 70.02 | 69.52 | 69.07 | 68.50 | | | 512 | 656 | 70.82 | 84.31 | 87.55 | 65.14 | 63.57 | 62.62 | 61.73 | 70.12 | 69.53 | 69.04 | 68.45 | | | 1024 | 1312 | 70.89 | 84.44 | 87.68 | 65.16 | 63.58 | 62.60 | 61.68 | 70.14 | 69.54 | 69.01 | 68.41 | | | 2048 | 2624 | 70.97 | 84.41 | 87.74 | 65.20 | 63.57 | 62.56 | 61.60 | 70.18 | 69.52 | 68.98 | 68.35 | | | ${\rm MRL}$ Interpolated | 12 | 15 | 65.89 | 80.04 | 83.68 | 60.84 | 59.66 | 58.98 | 58.37 | 65.94 | 65.72 | 65.45 | 65.08 | | 24 | 31 | 68.76 | 82.48 | 85.87 | 63.64 | 62.42 | 61.74 | 61.13 | 68.64 | 68.35 | 68.07 | 67.71 | | | 48 | 61 | 69.96 | 83.40 | 86.65 | 64.58 | 63.2 | 62.42 | 61.72 | 69.53 | 69.10 | 68.75 | 68.32 | | | 96 | 123 | 70.40 | 83.83 | 87.04 | 64.86 | 63.46 | 62.62 | 61.84 | 69.82 | 69.38 | 68.98 | 68.48 | | | 192 | 246 | 70.64 | 84.09 | 87.37 | 65.00 | 63.53 | 62.66 | 61.83 | 69.98 | 69.49 | 69.05 | 68.50 | | | 384 | 492 | 70.69 | 84.25 | 87.41 | 65.09 | 63.56 | 62.64 | 61.76 | 70.05 | 69.51 | 69.04 | 68.46 | | | 768 | 984 | 70.84 | 84.40 | 87.63 | 65.16 | 63.59 | 62.62 | 61.71 | 70.14 | 69.55 | 69.03 | 68.44 | | | 1536 | 1968 | 70.88 | 84.39 | 87.71 | 65.18 | 63.59 | 62.58 | 61.64 | 70.16 | 69.54 | 68.99 | 68.38 | | Table 9: Retrieve a shortlist of 200-NN with $D_{s}$ sized representations on ImageNetV2 via exact search with L2 distance metric. Top-1 and mAP@10 entries (%) where ${\rm MRL\text{--}E}$ outperforms FF are bolded. ${\rm MRL}$ outperforms FF at all $D_{s}$ and is thus not bolded. | FF | 8 | 10 | 48.79 | 64.70 | 69.72 | 43.04 | 41.89 | 41.42 | 41.17 | 48.43 | 48.27 | 48.25 | 48.19 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 16 | 20 | 55.08 | 69.50 | 74.08 | 49.63 | 48.53 | 48.06 | 47.75 | 54.76 | 54.64 | 54.53 | 54.39 | | | 32 | 41 | 56.69 | 71.10 | 76.47 | 51.11 | 49.85 | 49.17 | 48.65 | 56.23 | 55.96 | 55.71 | 55.42 | | | 64 | 82 | 57.37 | 72.71 | 77.48 | 51.28 | 49.75 | 48.85 | 47.99 | 56.65 | 56.14 | 55.71 | 55.15 | | | 128 | 164 | 57.17 | 73.31 | 78.64 | 50.07 | 48.09 | 46.79 | 45.58 | 55.75 | 54.89 | 54.12 | 53.28 | | | 256 | 328 | 57.09 | 74.04 | 79.24 | 49.11 | 46.66 | 44.99 | 43.35 | 55.02 | 53.77 | 52.74 | 51.53 | | | 512 | 656 | 57.12 | 73.91 | 79.32 | 48.95 | 46.25 | 44.37 | 42.42 | 54.88 | 53.49 | 52.29 | 50.83 | | | 1024 | 1312 | 57.53 | 74.17 | 79.55 | 48.27 | 45.41 | 43.36 | 41.26 | 54.31 | 52.84 | 51.49 | 49.87 | | | 2048 | 2624 | 57.84 | 74.59 | 79.45 | 49.99 | 47.47 | 45.66 | 43.87 | 55.89 | 54.63 | 53.45 | 52.12 | | | ${\rm MRL\text{--}E}$ | 8 | 10 | 47.05 | 62.53 | 67.60 | 40.79 | 39.47 | 38.78 | 38.16 | 46.03 | 45.77 | 45.54 | 45.17 | | 16 | 20 | 55.73 | 70.54 | 74.86 | 49.86 | 48.57 | 47.84 | 47.26 | 54.97 | 54.71 | 54.44 | 54.10 | | | 32 | 41 | 57.33 | 71.61 | 76.64 | 51.26 | 49.92 | 49.09 | 48.42 | 56.46 | 56.11 | 55.70 | 55.30 | | | 64 | 82 | 57.90 | 72.55 | 77.44 | 51.89 | 50.29 | 49.34 | 48.53 | 57.06 | 56.45 | 55.97 | 55.43 | | | 128 | 164 | 57.73 | 72.79 | 77.28 | 52.02 | 50.38 | 49.49 | 48.62 | 57.13 | 56.58 | 56.15 | 55.58 | | | 256 | 328 | 58.22 | 72.77 | 77.67 | 52.16 | 50.61 | 49.67 | 48.81 | 57.30 | 56.79 | 56.33 | 55.77 | | | 512 | 656 | 58.46 | 73.00 | 77.88 | 52.52 | 50.97 | 50.02 | 49.16 | 57.65 | 57.10 | 56.64 | 56.08 | | | 1024 | 1312 | 58.71 | 73.29 | 78.00 | 52.70 | 51.13 | 50.17 | 49.30 | 57.83 | 57.26 | 56.77 | 56.20 | | | 2048 | 2624 | 58.86 | 73.17 | 78.00 | 52.88 | 51.25 | 50.26 | 49.36 | 57.95 | 57.35 | 56.85 | 56.25 | | | ${\rm MRL}$ | 8 | 10 | 50.41 | 65.56 | 70.27 | 45.51 | 44.38 | 43.71 | 43.17 | 50.55 | 50.44 | 50.17 | 49.91 | | 16 | 20 | 56.64 | 70.19 | 74.61 | 50.98 | 49.76 | 49.16 | 48.69 | 55.90 | 55.66 | 55.52 | 55.29 | | | 32 | 41 | 57.96 | 71.88 | 76.41 | 52.06 | 50.78 | 50.09 | 49.54 | 57.18 | 56.83 | 56.57 | 56.27 | | | 64 | 82 | 58.94 | 72.74 | 77.17 | 52.65 | 51.24 | 50.44 | 49.76 | 57.72 | 57.29 | 56.94 | 56.52 | | | 128 | 164 | 59.13 | 73.07 | 77.49 | 52.94 | 51.42 | 50.53 | 49.74 | 58.00 | 57.47 | 57.05 | 56.55 | | | 256 | 328 | 59.18 | 73.64 | 77.75 | 52.96 | 51.45 | 50.52 | 49.70 | 58.01 | 57.53 | 57.06 | 56.54 | | | 512 | 656 | 59.40 | 73.85 | 77.97 | 53.01 | 51.39 | 50.46 | 49.61 | 58.11 | 57.49 | 57.04 | 56.48 | | | 1024 | 1312 | 59.11 | 73.77 | 77.92 | 52.98 | 51.37 | 50.40 | 49.54 | 58.13 | 57.51 | 57.00 | 56.45 | | | 2048 | 2624 | 59.63 | 73.84 | 77.97 | 52.96 | 51.34 | 50.34 | 49.44 | 58.07 | 57.48 | 56.95 | 56.36 | | Table 10: Retrieve a shortlist of 200-NN with $D_{s}$ sized representations on ImageNet-4K via exact search with L2 distance metric. ${\rm MRL\text{--}E}$ and FF models are omitted for clarity and compute/inference time costs. All entries are in %. | Config | $D_{s}$ | MFLOPs | Top-1 | Top-5 | Top-10 | mAP@10 | mAP@25 | mAP@50 | mAP@100 | P@10 | P@25 | P@50 | P@100 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | ${\rm MRL}$ | 8 | 34 | 10.60 | 26.23 | 35.57 | 5.32 | 4.29 | 3.76 | 3.36 | 9.13 | 8.77 | 8.46 | 8.13 | | 16 | 67 | 16.74 | 36.91 | 47.28 | 8.64 | 6.83 | 5.84 | 5.05 | 13.82 | 12.79 | 12.04 | 13.27 | | | 32 | 134 | 21.54 | 43.75 | 54.11 | 11.36 | 8.88 | 7.47 | 6.31 | 17.25 | 15.67 | 14.47 | 13.27 | | | 64 | 269 | 25.00 | 47.97 | 58.25 | 13.38 | 10.40 | 8.67 | 7.23 | 19.68 | 17.64 | 16.14 | 14.65 | | | 128 | 538 | 27.27 | 50.35 | 60.47 | 14.77 | 11.47 | 9.53 | 7.91 | 21.25 | 18.95 | 17.26 | 15.59 | | | 256 | 1076 | 28.53 | 51.95 | 61.90 | 15.66 | 12.19 | 10.12 | 8.38 | 22.28 | 19.81 | 18.01 | 16.22 | | | 512 | 2151 | 29.46 | 53.03 | 62.81 | 16.29 | 12.70 | 10.55 | 8.72 | 22.96 | 20.42 | 18.54 | 16.68 | | | 1024 | 4303 | 30.23 | 53.72 | 63.45 | 16.76 | 13.08 | 10.86 | 8.97 | 23.48 | 20.88 | 18.93 | 17.00 | | | 2048 | 8606 | 30.87 | 54.32 | 64.02 | 17.20 | 13.43 | 11.14 | 9.19 | 23.97 | 21.28 | 19.28 | 17.30 | | | ${\rm MRL}$ - Interpolated | 12 | 50 | 14.04 | 32.56 | 42.71 | 7.16 | 5.70 | 4.92 | 4.32 | 11.81 | 11.08 | 10.52 | 9.94 | | 24 | 101 | 19.49 | 40.82 | 51.26 | 10.17 | 7.98 | 6.75 | 5.75 | 15.76 | 14.43 | 13.42 | 12.40 | | | 48 | 202 | 23.51 | 46.23 | 56.56 | 12.49 | 9.72 | 8.13 | 6.81 | 18.62 | 16.75 | 15.39 | 14.04 | | | 96 | 403 | 26.25 | 49.32 | 59.48 | 14.15 | 11.00 | 9.15 | 7.61 | 20.55 | 18.36 | 16.78 | 15.17 | | | 192 | 807 | 27.94 | 51.32 | 61.32 | 15.29 | 11.89 | 9.88 | 8.18 | 21.86 | 19.46 | 17.71 | 15.96 | | | 384 | 1614 | 29.03 | 52.53 | 62.45 | 15.99 | 12.46 | 10.35 | 8.56 | 22.64 | 20.14 | 18.29 | 16.47 | | | 768 | 3227 | 29.87 | 53.36 | 63.13 | 16.54 | 12.90 | 10.71 | 8.85 | 23.23 | 20.67 | 18.75 | 16.85 | | | 1536 | 6454 | 30.52 | 54.02 | 63.79 | 16.99 | 13.27 | 11.01 | 9.08 | 23.73 | 21.09 | 19.12 | 17.16 | | Retrieval performance on ImageNet-1K, i.e. the training distribution, is shown in Table 8. ${\rm MRL}$ outperforms FF models for nearly all representation size for both top-1 and mAP@10, and especially at low representation size ( $D_{s}$ $\leq 32$ ). ${\rm MRL\text{--}E}$ loses out to FF significantly only at $D_{s}$ $=8$ . This indicates that training ResNet50 models via the ${\rm MRL}$ training paradigm improves retrieval at low representation size over models explicitly trained at those representation size (FF- $8...2048$ ). We carried out all retrieval experiments at $D_{s}$ $\in\{8,16,32,64,128,256,512,1024,2048\}$ , as these were the representation sizes which were a part of the nesting_list at which losses were added during training, as seen in Algorithm 1, Appendix A. To examine whether ${\rm MRL}$ is able to learn ${\rm Matryoshka~Representations}$ at dimensions in between the representation size for which it was trained, we also tabulate the performance of ${\rm MRL}$ at interpolated $D_{s}$ $\in\{12,24,48,96,192,384,768,1536\}$ as ${\rm MRL}$ –Interpolated and ${\rm MRL\text{--}E}$ –Interpolated (see Table 8). We observed that performance scaled nearly monotonically between the original representation size and the interpolated representation size as we increase $D_{s}$ , which demonstrates that ${\rm MRL}$ is able to learn ${\rm Matryoshka~Representations}$ at nearly all representation size $m\in[8,2048]$ despite optimizing only for $|\mathcal{M}|$ nested representation sizes. We examined the robustness of ${\rm MRL}$ for retrieval on out-of-domain datasets ImageNetV2 and ImageNet-4K, as shown in Table 9 and Table 10 respectively. On ImageNetV2, we observed that ${\rm MRL}$ outperformed FF at all $D_{s}$ on top-1 Accuracy and mAP@10, and ${\rm MRL\text{--}E}$ outperformed FF at all $D_{s}$ except $D_{s}$ $=8$ . This demonstrates the robustness of the learned ${\rm Matryoshka~Representations}$ for out-of-domain image retrieval. ## Appendix F Adaptive Retrieval The time complexity of retrieving a shortlist of k-NN often scales as $O(d)$ , where $d=$ $D_{s}$ , for a fixed k and $N$ . We thus will have a theoretical $256\times$ higher cost for $D_{s}$ $=2048$ over $D_{s}$ $=8$ . We discuss search complexity in more detail in Appendix I. In an attempt to replicate performance at higher $D_{s}$ while using less FLOPs, we perform adaptive retrieval via retrieving a k-NN shortlist with representation size $D_{s}$ , and then re-ranking the shortlist with representations of size $D_{r}$ . Adaptive retrieval for a shortlist length $k=200$ is shown in Table 11 for ImageNet-1K, and in Table 12 for ImageNet-4K. On ImageNet-1K, we are able to achieve comparable performance to retrieval with $D_{s}$ $=2048$ (from Table 8) with $D_{s}$ $=16$ at $128\times$ less MFLOPs/Query (used interchangeably with MFLOPs). Similarly, on ImageNet-4K, we are able to achieve comparable performance to retrieval with $D_{s}$ $=2048$ (from Table 10) with $D_{s}$ $=64$ on ImageNet-1K and ImageNet-4K, at $32\times$ less MFLOPs. This demonstrates the value of intelligent routing techniques which utilize appropriately sized ${\rm Matryoshka~Representations}$ for retrieval. Table 11: Retrieve a shortlist of k-NN with $D_{s}$ sized representations on ImageNet-1K with ${\rm MRL}$ representations, and then re-order the neighbors shortlist with L2 distances using $D_{r}$ sized representations. Top-1 and mAP@10 entries (%) that are within $0.1\$ of the maximum value achievable without reranking on ${\rm MRL}$ representations, as seen in Table 8, are bolded. | Shortlist Length = 200 | $D_{s}$ | $D_{r}$ | MFLOPs | Top-1 | mAP@10 | mAP@25 | mAP@50 | mAP@100 | P@10 | P@25 | P@50 | P@100 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 8 | 16 | 10 | 68.21 | 63.35 | 62.25 | 61.70 | 61.19 | 68.32 | 68.14 | 67.96 | 67.65 | | | 32 | 69.42 | 64.12 | 62.81 | 62.03 | 61.32 | 69.04 | 68.63 | 68.22 | 67.71 | | | | | 64 | 70.05 | 64.46 | 63.03 | 62.14 | 61.29 | 69.37 | 68.83 | 68.32 | 67.66 | | | | | 128 | 70.34 | 64.68 | 63.16 | 62.21 | 61.27 | 69.59 | 68.96 | 68.38 | 67.65 | | | | | 256 | 70.40 | 64.77 | 63.21 | 62.23 | 61.26 | 69.66 | 69.02 | 68.41 | 67.65 | | | | | 512 | 70.60 | 64.86 | 63.22 | 62.21 | 61.22 | 69.74 | 69.02 | 68.39 | 67.62 | | | | | 1024 | 70.71 | 64.88 | 63.23 | 62.20 | 61.20 | 69.76 | 69.01 | 68.39 | 67.60 | | | | | 2048 | 70.81 | 64.90 | 63.22 | 62.17 | 61.16 | 69.77 | 68.99 | 68.36 | 67.57 | | | | | 16 | 32 | 21 | 69.47 | 64.27 | 63.04 | 62.36 | 61.75 | 69.21 | 68.90 | 68.58 | 68.12 | | | 64 | 70.16 | 64.74 | 63.42 | 62.66 | 61.94 | 69.66 | 69.22 | 68.81 | 68.22 | | | | | 128 | 70.52 | 65.00 | 63.60 | 62.77 | 61.98 | 69.91 | 69.36 | 68.89 | 68.24 | | | | | 256 | 70.55 | 65.10 | 63.67 | 62.82 | 62.01 | 69.98 | 69.43 | 68.92 | 68.25 | | | | | 512 | 70.74 | 65.21 | 63.70 | 62.83 | 62.00 | 70.08 | 69.43 | 68.92 | 68.24 | | | | | 1024 | 70.83 | 65.23 | 63.72 | 62.83 | 61.99 | 70.08 | 69.45 | 68.92 | 68.23 | | | | | 2048 | 70.90 | 65.27 | 63.73 | 62.82 | 61.97 | 70.10 | 69.44 | 68.90 | 68.21 | | | | | 32 | 64 | 41 | 70.16 | 64.69 | 63.35 | 62.57 | 61.93 | 69.68 | 69.26 | 68.92 | 68.51 | | | 128 | 70.52 | 64.97 | 63.54 | 62.73 | 62.04 | 69.95 | 69.47 | 69.06 | 68.59 | | | | | 256 | 70.63 | 65.07 | 63.63 | 62.79 | 62.07 | 70.04 | 69.55 | 69.12 | 68.61 | | | | | 512 | 70.82 | 65.17 | 63.66 | 62.80 | 62.06 | 70.11 | 69.57 | 69.12 | 68.60 | | | | | 1024 | 70.89 | 65.20 | 63.68 | 62.80 | 62.04 | 70.15 | 69.59 | 69.12 | 68.59 | | | | | 2048 | 70.97 | 65.24 | 63.70 | 62.79 | 62.02 | 70.19 | 69.59 | 69.10 | 68.56 | | | | | 64 | 128 | 82 | 70.51 | 64.94 | 63.50 | 62.64 | 61.88 | 69.94 | 69.44 | 69.02 | 68.54 | | | 256 | 70.63 | 65.04 | 63.57 | 62.69 | 61.91 | 70.02 | 69.52 | 69.08 | 68.57 | | | | | 512 | 70.83 | 65.14 | 63.59 | 62.67 | 61.87 | 70.12 | 69.54 | 69.06 | 68.54 | | | | | 1024 | 70.89 | 65.16 | 63.59 | 62.65 | 61.85 | 70.15 | 69.54 | 69.05 | 68.52 | | | | | 2048 | 70.97 | 65.20 | 63.59 | 62.63 | 61.82 | 70.18 | 69.53 | 69.03 | 68.49 | | | | | 128 | 256 | 164 | 70.63 | 65.04 | 63.56 | 62.66 | 61.82 | 70.02 | 69.52 | 69.07 | 68.51 | | | 512 | 70.82 | 65.14 | 63.58 | 62.63 | 61.77 | 70.11 | 69.54 | 69.04 | 68.47 | | | | | 1024 | 70.89 | 65.16 | 63.58 | 62.60 | 61.73 | 70.14 | 69.54 | 69.02 | 68.45 | | | | | 2048 | 70.97 | 65.20 | 63.57 | 62.57 | 61.68 | 70.18 | 69.52 | 68.99 | 68.41 | | | | | 256 | 512 | 328 | 70.82 | 65.14 | 63.57 | 62.62 | 61.74 | 70.12 | 69.53 | 69.04 | 68.45 | | | 1024 | 70.88 | 65.16 | 63.58 | 62.60 | 61.69 | 70.14 | 69.54 | 69.01 | 68.41 | | | | | 2048 | 70.97 | 65.20 | 63.56 | 62.56 | 61.62 | 70.18 | 69.52 | 68.98 | 68.37 | | | | | 512 | 1024 | 656 | 70.90 | 65.16 | 63.58 | 62.60 | 61.68 | 70.14 | 69.54 | 69.01 | 68.41 | | | 2048 | 70.98 | 65.20 | 63.57 | 62.56 | 61.60 | 70.18 | 69.52 | 68.98 | 68.35 | | | | | 1024 | 2048 | 1312 | 70.97 | 65.20 | 63.57 | 62.56 | 61.60 | 70.18 | 69.52 | 68.98 | 68.35 | | Table 12: Retrieve a shortlist of k-NN with $D_{s}$ sized representations on ImageNet-4K with ${\rm MRL}$ representations, and then re-order the neighbors shortlist with L2 distances using $D_{r}$ sized representations. Top-1 and mAP@10 entries (%) that are within $0.1\$ of the maximum value achievable without reranking on ${\rm MRL}$ representations, as seen in Table 10, are bolded. | Shortlist Length = 200 | $D_{s}$ | $D_{r}$ | MFLOPs | Top-1 | mAP@10 | mAP@25 | mAP@50 | mAP@100 | P@10 | P@25 | P@50 | P@100 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 8 | 16 | 34 | 16.84 | 8.70 | 6.88 | 5.88 | 5.08 | 13.86 | 12.80 | 11.98 | 11.10 | | | 32 | 20.73 | 10.66 | 8.19 | 6.77 | 5.61 | 16.18 | 14.39 | 13.02 | 11.61 | | | | | 64 | 23.11 | 11.91 | 9.03 | 7.36 | 6.00 | 17.56 | 15.34 | 13.67 | 11.99 | | | | | 128 | 24.63 | 12.71 | 9.59 | 7.76 | 6.25 | 18.42 | 15.94 | 14.08 | 12.22 | | | | | 256 | 25.5 | 13.24 | 9.96 | 8.03 | 6.42 | 19.00 | 16.35 | 14.36 | 12.37 | | | | | 512 | 26.07 | 13.59 | 10.21 | 8.20 | 6.53 | 19.37 | 16.62 | 14.54 | 12.46 | | | | | 1024 | 26.52 | 13.85 | 10.40 | 8.34 | 6.61 | 19.65 | 16.80 | 14.68 | 12.53 | | | | | 2048 | 26.94 | 14.11 | 10.57 | 8.45 | 6.68 | 19.92 | 16.98 | 14.79 | 12.58 | | | | | 16 | 32 | 67 | 21.44 | 11.24 | 8.72 | 7.26 | 6.02 | 17.02 | 15.30 | 13.92 | 12.41 | | | 64 | 24.36 | 12.78 | 9.75 | 7.96 | 6.43 | 18.72 | 16.41 | 14.63 | 12.74 | | | | | 128 | 26.08 | 13.70 | 10.39 | 8.39 | 6.69 | 19.68 | 17.07 | 15.05 | 12.94 | | | | | 256 | 26.99 | 14.27 | 10.79 | 8.67 | 6.85 | 20.27 | 17.48 | 15.31 | 13.07 | | | | | 512 | 27.60 | 14.66 | 11.06 | 8.86 | 6.97 | 20.67 | 17.75 | 15.50 | 13.16 | | | | | 1024 | 28.12 | 14.94 | 11.26 | 8.99 | 7.05 | 20.96 | 17.95 | 15.62 | 13.22 | | | | | 2048 | 28.56 | 15.21 | 11.43 | 9.11 | 7.12 | 21.23 | 18.13 | 15.73 | 13.27 | | | | | 32 | 64 | 134 | 24.99 | 13.35 | 10.35 | 8.59 | 7.09 | 19.61 | 17.52 | 15.92 | 14.21 | | | 128 | 27.17 | 14.61 | 11.27 | 9.26 | 7.51 | 20.99 | 18.52 | 16.62 | 14.59 | | | | | 256 | 28.33 | 15.37 | 11.83 | 9.67 | 7.77 | 21.80 | 19.12 | 17.05 | 14.81 | | | | | 512 | 29.12 | 15.88 | 12.20 | 9.94 | 7.93 | 22.33 | 19.51 | 17.32 | 14.94 | | | | | 1024 | 29.78 | 16.25 | 12.47 | 10.13 | 8.05 | 22.71 | 19.79 | 17.5 | 15.03 | | | | | 2048 | 30.33 | 16.59 | 12.72 | 10.30 | 8.16 | 23.07 | 20.05 | 17.66 | 15.11 | | | | | 64 | 128 | 269 | 27.27 | 14.76 | 11.47 | 9.51 | 7.85 | 21.25 | 18.92 | 17.20 | 15.40 | | | 256 | 28.54 | 15.64 | 12.15 | 10.05 | 8.21 | 22.24 | 19.71 | 17.81 | 15.76 | | | | | 512 | 29.45 | 16.25 | 12.62 | 10.40 | 8.44 | 22.88 | 20.24 | 18.20 | 15.97 | | | | | 1024 | 30.19 | 16.69 | 12.96 | 10.66 | 8.60 | 23.35 | 20.61 | 18.46 | 16.10 | | | | | 2048 | 30.81 | 17.10 | 13.27 | 10.88 | 8.74 | 23.79 | 20.93 | 18.69 | 16.21 | | | | | 128 | 256 | 538 | 28.54 | 15.66 | 12.19 | 10.12 | 8.36 | 22.28 | 19.81 | 18.00 | 16.16 | | | 512 | 29.45 | 16.29 | 12.69 | 10.53 | 8.66 | 22.96 | 20.41 | 18.50 | 16.48 | | | | | 1024 | 30.22 | 16.76 | 13.07 | 10.83 | 8.86 | 23.47 | 20.84 | 18.83 | 16.68 | | | | | 2048 | 30.86 | 17.19 | 13.41 | 11.09 | 9.03 | 23.95 | 21.22 | 19.12 | 16.84 | | | | | 256 | 512 | 1076 | 29.45 | 16.29 | 12.70 | 10.55 | 8.71 | 22.97 | 20.42 | 18.54 | 16.66 | | | 1024 | 30.21 | 16.76 | 13.08 | 10.86 | 8.95 | 23.48 | 20.87 | 18.92 | 16.94 | | | | | 2048 | 30.85 | 17.20 | 13.43 | 11.14 | 9.15 | 23.97 | 21.27 | 19.26 | 17.16 | | | | | 512 | 1024 | 2152 | 30.22 | 16.76 | 13.08 | 10.86 | 8.97 | 23.48 | 20.88 | 18.93 | 17.00 | | | 2048 | 30.87 | 17.20 | 13.43 | 11.14 | 9.19 | 23.97 | 21.28 | 19.28 | 17.28 | | | | | 1024 | 2048 | 4303 | 30.87 | 17.20 | 13.43 | 11.15 | 9.19 | 23.97 | 21.28 | 19.28 | 17.29 | | Funnel Retrieval. We also designed a simple cascade policy which we call funnel retrieval to successively improve and refine the k-NN shortlist at increasing $D_{s}$ . This was an attempt to remove the dependence on manual choice of $D_{s}$ & $D_{r}$ . We retrieved a shortlist at $D_{s}$ and then re-ranked the shortlist five times while simultaneously increasing $D_{r}$ (rerank cascade) and decreasing the shortlist length (shortlist cascade), which resembles a funnel structure. We tabulate the performance of funnel retrieval in various configurations in Table 13 on ImageNet-1K, and in Table 14 on ImageNet-4K. With funnel retrieval on ImageNet-1K, we were able to achieve top-1 accuracy within $0.1\$ of retrieval with $D_{s}$ $=2048$ (as in Table 8) with a funnel with $D_{s}$ $=16$ , with $128\times$ less MFLOPs. Similarly, we are able to achieve equivalent top-1 accuracy within $0.15\$ of retrieval at $D_{s}$ $=2048$ (as in Table 10) with funnel retrieval at $D_{s}$ $=32$ on ImageNet-4K, with $64\times$ less MFLOPs. This demonstrates that with funnel retrieval, we can emulate the performance of retrieval with $D_{s}$ $=2048$ with a fraction of the MFLOPs. Table 13: Retrieve a shortlist of k-NN with $D_{s}$ sized representations on ImageNet-1K with ${\rm MRL}$ . This shortlist is then reranked with funnel retrieval, which uses a rerank cascade with a one-to-one mapping with a monotonically decreasing shortlist length as shown in the shortlist cascade. Top-1 and mAP@10 entries (%) within $0.1\$ of the maximum achievable without reranking on ${\rm MRL}$ representations, as seen in Table 8, are bolded. | 8 | 16 $\to$ 32 $\to$ 64 $\to$ 128 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 10.28 | 70.22 | 82.63 | 85.49 | 64.06 | 68.65 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 10.29 | 70.46 | 83.13 | 86.08 | 64.43 | 69.10 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 10.31 | 70.58 | 83.54 | 86.53 | 64.62 | 69.37 | | | | 16 | 32 $\to$ 64 $\to$ 128 $\to$ 256 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 20.54 | 70.90 | 83.96 | 86.85 | 65.19 | 69.97 | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 20.56 | 70.95 | 84.05 | 87.04 | 65.18 | 70.00 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 20.61 | 70.96 | 84.18 | 87.22 | 65.14 | 70.01 | | | | 32 | 64 $\to$ 128 $\to$ 256 $\to$ 512 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 41.07 | 70.96 | 84.32 | 87.47 | 65.21 | 70.11 | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 41.09 | 70.97 | 84.32 | 87.47 | 65.19 | 70.11 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 41.20 | 70.97 | 84.36 | 87.53 | 65.18 | 70.11 | | | Table 14: Retrieve a shortlist of k-NN with $D_{s}$ sized representations on ImageNet-4K with ${\rm MRL}$ . This shortlist is then reranked with funnel retrieval, which uses a rerank cascade with a one-to-one mapping with a monotonically decreasing shortlist length as shown in the shortlist cascade. Top-1 and mAP@10 entries (%) within $0.15\$ of the maximum achievable without reranking on ${\rm MRL}$ representations, as seen in Table 10, are bolded. | 8 | 16 $\to$ 32 $\to$ 64 $\to$ 128 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 33.65 | 26.20 | 46.45 | 54.12 | 12.79 | 17.85 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 33.66 | 26.55 | 47.02 | 54.72 | 13.02 | 18.15 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 33.68 | 26.83 | 47.54 | 55.35 | 13.24 | 18.44 | | | | 16 | 32 $\to$ 64 $\to$ 128 $\to$ 256 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 67.28 | 29.51 | 51.44 | 59.56 | 15.27 | 21.03 | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 67.29 | 29.66 | 51.71 | 59.88 | 15.42 | 21.22 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 67.34 | 29.79 | 52.00 | 60.25 | 15.55 | 21.41 | | | | 32 | 64 $\to$ 128 $\to$ 256 $\to$ 512 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 134.54 | 30.64 | 53.52 | 62.16 | 16.45 | 22.64 | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 134.56 | 30.69 | 53.65 | 62.31 | 16.51 | 22.73 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 134.66 | 30.72 | 53.78 | 62.43 | 16.55 | 22.79 | | | | 64 | 128 $\to$ 256 $\to$ 512 $\to$ 1024 $\to$ 2048 | 200 $\to$ 100 $\to$ 50 $\to$ 25 $\to$ 10 | 269.05 | 30.81 | 54.06 | 63.15 | 16.87 | 23.34 | | 400 $\to$ 200 $\to$ 50 $\to$ 25 $\to$ 10 | 269.10 | 30.84 | 54.20 | 63.31 | 16.92 | 23.42 | | | | 800 $\to$ 400 $\to$ 200 $\to$ 50 $\to$ 10 | 269.31 | 30.87 | 54.27 | 63.42 | 16.95 | 23.46 | | | ## Appendix G Few-shot and Sample Efficiency We compared ${\rm MRL}$ , ${\rm MRL\text{--}E}$ , and FF on various benchmarks to observe the effect of representation size on sample efficiency. We used Nearest Class Means [79] for classification which has been shown to be effective in the few-shot regime [13]. ImageNetV2. Representations are evaluated on ImageNetV2 with the n-shot k-way setup. ImageNetV2 is a dataset traditionally used to evaluate the robustness of models to natural distribution shifts. For our experiments we evaluate accuracy of the model given $n$ examples from the ImageNetV2 distribution. We benchmark representations in the traditional small-scale (10-way) and large-scale (1000-way) setting. We evaluate for $n\in{1,3,5,7,9}$ with 9 being the maximum value for $n$ because there are 10 images per class. We observed that ${\rm MRL}$ had equal performance to FF across all representation sizes and shot numbers. We also found that for both ${\rm MRL}$ and FF, as the shot number decreased, the required representation size to reach optimal accuracy decreased (Table 15). For example, we observed that 1-shot performance at $32$ representation size had equal accuracy to $2048$ representation size. Table 15: Few-shot accuracy (%) on ImageNetV2 for 1000-way classification. ${\rm MRL}$ performs equally to FF across all shots and representation sizes. We also observed that accuracy saturated at a lower dimension for lower shot numbers. E.g. for 1-shot, 32-dim performed comparably to 2048-dim. | 8 ${\rm MRL}$ 16 | FF 35.37 FF | 35.41 45.69 40.88 | 45.73 49.25 53.96 | 49.23 50.85 57.36 | 50.89 51.73 58.72 | 51.72 59.39 | | --- | --- | --- | --- | --- | --- | --- | | ${\rm MRL}$ | 40.90 | 53.94 | 57.37 | 58.65 | 59.29 | | | 32 | FF | 41.41 | 54.88 | 58.28 | 59.63 | 60.40 | | ${\rm MRL}$ | 41.40 | 54.91 | 58.30 | 59.65 | 60.45 | | | 64 | FF | 41.25 | 54.83 | 58.29 | 59.82 | 60.61 | | ${\rm MRL}$ | 41.28 | 54.80 | 58.32 | 59.77 | 60.69 | | | 128 | FF | 41.36 | 54.90 | 58.50 | 60.05 | 60.90 | | ${\rm MRL}$ | 41.38 | 54.95 | 58.50 | 60.06 | 60.83 | | | 256 | FF | 41.36 | 54.90 | 58.50 | 60.05 | 60.90 | | ${\rm MRL}$ | 41.38 | 54.95 | 58.50 | 60.06 | 60.83 | | | 512 | FF | 41.36 | 55.05 | 58.70 | 60.19 | 61.02 | | ${\rm MRL}$ | 41.34 | 55.14 | 58.78 | 60.40 | 61.18 | | | 1024 | FF | 41.32 | 55.20 | 58.85 | 60.46 | 61.38 | | ${\rm MRL}$ | 41.31 | 55.24 | 58.86 | 60.42 | 61.34 | | | 2048 | FF | 41.18 | 55.09 | 58.77 | 60.38 | 61.34 | | ${\rm MRL}$ | 41.16 | 55.10 | 58.77 | 60.40 | 61.28 | | FLUID. For the long-tailed setting we evaluated ${\rm MRL}$ on the FLUID benchmark [92] which contains a mixture of pretrain and new classes. Table 16 shows the evaluation of the learned representation on FLUID. We observed that ${\rm MRL}$ provided up to 2% higher accuracy on novel classes in the tail of the distribution, without sacrificing accuracy on other classes. Additionally we found the accuracy between low-dimensional and high-dimensional representations was marginal for pretrain classes. For example, the 64-dimensional ${\rm MRL}$ performed $\sim 1\$ lower in accuracy compared to the 2048-dimensional counterpart on pretrain-head classes (84.46% vs 85.60%). However for novel-tail classes the gap was far larger (6.22% vs 12.88%). We hypothesize that the higher-dimensional representations are required to differentiate the classes when few training examples of each are known. These results provide further evidence that different tasks require varying capacity based on their difficulty. Table 16: Accuracy (%) categories indicates whether classes were present during ImageNet pretraining and head/tail indicates classes that have greater/less than 50 examples in the streaming test set. We observed that ${\rm MRL}$ performed better than the baseline on novel tail classes by $\sim 2\$ on average. | 8 ${\rm MRL}$ ${\rm MRL\text{--}E}$ | FF 71.75 57.40 | 68.04 10.70 6.25 | 11.30 38.29 23.14 | 33.18 0.19 0.04 | 0.36 17.15 11.78 | 16.29 29.34 22.81 | 28.47 | | --- | --- | --- | --- | --- | --- | --- | --- | | 16 | FF | 80.74 | 19.12 | 63.29 | 2.78 | 25.65 | 37.61 | | ${\rm MRL}$ | 81.79 | 17.90 | 61.39 | 1.95 | 24.73 | 37.59 | | | ${\rm MRL\text{--}E}$ | 79.08 | 9.15 | 60.33 | 0.08 | 20.45 | 30.24 | | | 32 | FF | 83.67 | 24.30 | 66.66 | 4.23 | 28.86 | 42.40 | | ${\rm MRL}$ | 83.46 | 23.26 | 65.82 | 3.75 | 28.16 | 41.90 | | | ${\rm MRL\text{--}E}$ | 81.42 | 10.47 | 68.01 | 0.23 | 22.31 | 32.17 | | | 64 | FF | 84.12 | 27.49 | 68.20 | 5.17 | 30.64 | 45.18 | | ${\rm MRL}$ | 84.46 | 27.61 | 67.59 | 6.22 | 31.03 | 45.35 | | | ${\rm MRL\text{--}E}$ | 82.57 | 13.23 | 70.18 | 0.52 | 23.83 | 34.74 | | | 128 | FF | 84.87 | 29.96 | 68.79 | 5.54 | 31.84 | 47.06 | | ${\rm MRL}$ | 84.88 | 30.86 | 68.58 | 8.41 | 33.23 | 47.79 | | | ${\rm MRL\text{--}E}$ | 82.76 | 18.93 | 64.46 | 2.22 | 25.75 | 39.19 | | | 256 | FF | 84.77 | 32.78 | 69.96 | 7.21 | 33.65 | 49.15 | | ${\rm MRL}$ | 85.10 | 32.91 | 69.39 | 9.99 | 34.74 | 49.39 | | | ${\rm MRL\text{--}E}$ | 82.96 | 22.63 | 64.55 | 3.59 | 27.64 | 41.96 | | | 512 | FF | 85.62 | 35.27 | 70.27 | 9.05 | 35.42 | 51.14 | | ${\rm MRL}$ | 85.62 | 34.67 | 70.24 | 11.43 | 36.11 | 50.79 | | | ${\rm MRL\text{--}E}$ | 82.86 | 25.62 | 64.34 | 4.99 | 29.22 | 44.20 | | | 1024 | FF | 86.30 | 37.49 | 71.12 | 10.92 | 37.14 | 52.88 | | ${\rm MRL}$ | 85.64 | 35.88 | 70.02 | 12.19 | 36.80 | 51.58 | | | ${\rm MRL\text{--}E}$ | 83.03 | 27.78 | 64.58 | 6.32 | 30.57 | 45.71 | | | 2048 | FF | 86.40 | 37.09 | 71.74 | 10.77 | 37.04 | 52.67 | | ${\rm MRL}$ | 85.60 | 36.83 | 70.34 | 12.88 | 37.46 | 52.18 | | | ${\rm MRL\text{--}E}$ | 83.01 | 29.99 | 65.37 | 7.60 | 31.97 | 47.16 | | ## Appendix H Robustness Experiments Table 17: Top-1 classification accuracy (%) on out-of-domain datasets (ImageNet-V2/R/A/Sketch) to examine robustness of ${\rm Matryoshka~Representation~Learning}$ . Note that these results are without any fine tuning on these datasets. | 8 16 32 | 65.86 73.10 74.68 | 56.92 72.38 74.80 | 67.46 73.80 75.26 | 54.05 60.52 62.24 | 47.40 60.48 62.23 | 55.59 61.71 63.05 | 24.60 28.51 31.28 | 22.98 28.45 30.79 | 23.57 28.85 31.47 | 2.92 3.00 2.60 | 3.63 3.55 3.65 | 3.39 3.59 3.57 | 17.73 21.70 22.03 | 15.07 20.38 21.87 | 17.98 21.77 22.48 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 64 | 75.45 | 75.48 | 76.17 | 63.51 | 63.15 | 63.99 | 32.96 | 32.13 | 33.39 | 2.87 | 3.99 | 3.76 | 22.13 | 22.56 | 23.43 | | 128 | 75.47 | 76.05 | 76.46 | 63.67 | 63.52 | 64.69 | 33.93 | 33.48 | 34.54 | 2.81 | 3.71 | 3.73 | 22.73 | 22.73 | 23.70 | | 256 | 75.78 | 76.31 | 76.66 | 64.13 | 63.80 | 64.71 | 34.80 | 33.91 | 34.85 | 2.77 | 3.65 | 3.60 | 22.63 | 22.88 | 23.59 | | 512 | 76.30 | 76.48 | 76.82 | 64.11 | 64.09 | 64.78 | 35.53 | 34.20 | 34.97 | 2.37 | 3.57 | 3.59 | 23.41 | 22.89 | 23.67 | | 1024 | 76.74 | 76.60 | 76.93 | 64.43 | 64.20 | 64.95 | 36.06 | 34.22 | 34.99 | 2.53 | 3.56 | 3.68 | 23.44 | 22.98 | 23.72 | | 2048 | 77.10 | 76.65 | 76.95 | 64.69 | 64.17 | 64.93 | 37.10 | 34.29 | 35.07 | 2.93 | 3.49 | 3.59 | 24.05 | 23.01 | 23.70 | Table 18: Zero-shot top-1 image classification accuracy (%) of a ALIGN- ${\rm MRL}$ model on ImageNet-V1/V2/R/A and ObjectNet. | 12 24 48 | 30.57 45.64 53.84 | 23.98 37.71 46.16 | 14.59 22.75 28.88 | 24.24 46.40 60.71 | 25.52 35.89 42.76 | | --- | --- | --- | --- | --- | --- | | 96 | 58.31 | 51.34 | 33.21 | 70.12 | 45.20 | | 192 | 60.95 | 53.56 | 36.10 | 74.41 | 48.24 | | 384 | 62.06 | 54.77 | 37.95 | 76.51 | 49.10 | | 768 | 62.26 | 55.15 | 37.84 | 76.73 | 49.26 | | Baseline | 66.39 | 59.57 | 39.97 | 80.49 | 51.60 | We evaluated the robustness of ${\rm MRL}$ models on out-of-domain datasets (ImageNetV2/R/A/Sketch) and compared them to the FF baseline. Each of these datasets is described in Appendix B. The results in Table 17 demonstrate that learning ${\rm Matryoshka~Representations}$ does not hurt out-of-domain generalization relative to FF models, and ${\rm Matryoshka~Representations}$ in fact improve the performance on ImageNet-A. For a ALIGN– ${\rm MRL}$ model, we examine the the robustness via zero-shot retrieval on out-of-domain datasets, including ObjectNet, in Table 18. ## Appendix I In Practice Costs All approximate NN search experiments via HNSW32 were run on an Intel Xeon 2.20GHz CPU with 24 cores. All exact search experiments were run with CUDA 11.0 on 2xA100-SXM4 NVIDIA GPUs with 40G RAM each. ${\rm MRL}$ models. As ${\rm MRL}$ makes minimal modifications to the ResNet50 model in the final fc layer via multiple heads for representations at various scales, it has only an 8MB storage overhead when compared to a standard ResNet50 model. ${\rm MRL\text{--}E}$ has no storage overhead as it has a shared head for logits at the final fc layer. Retrieval Exact search has a search time complexity of $O(dkN)$ , and HNSW has a search time complexity of $O(dk\log(N))$ , where $N$ is the database size, $d$ is the representation size, and $k$ is the shortlist length. To examine real-world performance, we tabulated wall clock search time for every query in the ImageNet-1K and ImageNet-4K validation sets over all representation sizes $d$ in Table 19 for both Exact Search and HNSW32, and ablated wall clock query time over shortlist length $k$ on the ImageNet-1K validation set in Table 21. The wall clock time to build the index and the index size is also shown in Table 20. Table 19: Retrieval k-NN wall clock search times (s) over the entire validation (query) set of ImageNet-1K and ImageNet-4K, containing 50K and 200K samples respectively. | 8 16 32 | 0.60 0.57 0.60 | 0.14 0.18 0.20 | 35.70 36.16 36.77 | 1.17 1.65 1.75 | | --- | --- | --- | --- | --- | | 64 | 0.66 | 0.24 | 27.88 | 2.21 | | 128 | 0.86 | 0.32 | 30.10 | 4.15 | | 256 | 1.29 | 0.46 | 34.97 | 3.39 | | 512 | 2.17 | 0.68 | 46.97 | 4.83 | | 1024 | 3.89 | 1.05 | 70.59 | 7.14 | | 2048 | 7.31 | 2.05 | 117.78 | 13.43 | Table 20: FAISS [47] index size and build times for exact k-NN search with L2 Distance metric and approximate k-NN search with HNSW32 [62]. | 8 16 32 | 40 80 160 | 0.04 0.08 0.16 | 131 263 525 | 0.33 0.27 0.52 | 381 421 501 | 4.87 6.15 6.80 | 1248 1379 1642 | 24.04 33.31 37.41 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 64 | 320 | 0.38 | 1051 | 1.05 | 661 | 8.31 | 2167 | 47.23 | | 128 | 641 | 0.64 | 2101 | 2.10 | 981 | 11.73 | 3218 | 89.87 | | 256 | 1281 | 1.27 | 4202 | 4.20 | 1622 | 17.70 | 5319 | 102.84 | | 512 | 2562 | 2.52 | 8404 | 8.39 | 2903 | 27.95 | 9521 | 158.47 | | 1024 | 5125 | 5.10 | 16808 | 17.20 | 5465 | 44.02 | 17925 | 236.30 | | 2048 | 10249 | 10.36 | 33616 | 41.05 | 10590 | 86.15 | 34733 | 468.18 | Table 21: Retrieval k-NN wall clock search times (s) over entire validation (query) set of ImageNet-1K over various shortlist lengths $k$ . | Exact L2 | 0.4406 | 0.4605 | 0.5736 | 0.6060 | 1.2781 | 2.7047 | | --- | --- | --- | --- | --- | --- | --- | | HNSW32 | 0.1193 | 0.1455 | 0.1833 | 0.2145 | 0.2333 | 0.2670 | ## Appendix J Analysis of Model Disagreement Class Trends Does increasing representation size necessarily help improve classification performance across all classes in ImageNet-1K? We studied this question by examining trends in performance with increasing representation size from $d={8,...2048}$ . For ${\rm MRL}$ models, we observed that $244$ classes showed a monotonic improvement in performance with increasing $d$ , $177$ classes first improved but then observed a slight dip (one or two misclassifications per class), $49$ classes showed a decline first and then an improvement, and the remaining classes did not show a clear trend. When we repeated this experiment with independently trained FF models, we noticed that $950$ classes did not show a clear trend. This motivated us to leverage the disagreement as well as gradual improvement of accuracy at different representation sizes by training ${\rm Matryoshka~Representations}$ . Figure 12 showcases the progression of relative per-class accuracy distribution compared to the ${\rm Matryoshka~Representation~Learning}$ -2048 dimensional model. This also showed that some instances and classes could benefit from lower-dimensional representations. Discussion of Oracle Accuracy Based on our observed model disagreements for different representation sizes $d$ , we defined an optimal oracle accuracy [58] for ${\rm MRL}$ . We labeled an image as correctly predicted if classification using any representation size was correct. The percentage of total samples of ImageNet-1K that were firstly correctly predicted using each representation size $d$ is shown in Table 22. This defined an upper bound on the performance of ${\rm MRL}$ models, as $18.46\$ of the ImageNet-1K validation set were incorrectly predicted $\forall d\in\{8,16,\ldots,2048\}$ . We show the oracle performance on ${\rm MRL}$ models for ImageNet-1K/V2/A/R/Sketch datasets in Table 23. <details> <summary>x26.png Details</summary>  ### Visual Description ## Bar Charts: Relative Performance Distribution Across Different 'd' Values ### Overview The image contains four bar charts comparing the distribution of relative performance (%) across four distinct 'd' values (8, 16, 64, 256). Each chart shows the frequency of classes (# Classes) at different relative performance levels, with a red "X" marker highlighting a specific data point in each distribution. ### Components/Axes - **X-axis**: Relative Performance (%) ranging from -60% to +20% in 20% increments. - **Y-axis**: Number of Classes (0–200) in 50-unit increments. - **Legends**: Each chart is labeled with its corresponding 'd' value (e.g., "d=8", "d=16", etc.). - **Red "X" Marker**: Positioned consistently at ~15% relative performance across all charts. ### Detailed Analysis 1. **d=8**: - Peak frequency at 0% relative performance (~180 classes). - Symmetric decline toward ±20%, with ~100 classes at -20% and ~20 classes at +20%. - Red "X" at +15% (~10 classes). 2. **d=16**: - Peak at 0% (~160 classes). - Asymmetric distribution: ~80 classes at -20%, ~40 classes at +20%. - Red "X" at +15% (~15 classes). 3. **d=64**: - Peak at 0% (~140 classes). - Narrower distribution: ~60 classes at -20%, ~30 classes at +20%. - Red "X" at +15% (~20 classes). 4. **d=256**: - Peak at 0% (~120 classes). - Further narrowed distribution: ~40 classes at -20%, ~25 classes at +20%. - Red "X" at +15% (~25 classes). ### Key Observations - **Symmetry**: All distributions are roughly symmetric around 0% relative performance, except for d=16, which shows slight asymmetry. - **Red "X" Consistency**: The red "X" marker is positioned identically at +15% across all charts, suggesting a standardized threshold or anomaly. - **Trend with 'd'**: As 'd' increases, the peak frequency at 0% decreases slightly, and the distribution narrows, indicating reduced variability in relative performance for higher 'd' values. ### Interpretation The data suggests that higher 'd' values correlate with more concentrated distributions of relative performance, centered at 0%. This could imply that larger 'd' values reduce variability in performance outcomes. The persistent red "X" at +15% across all charts may represent a critical performance benchmark or outlier threshold, warranting further investigation. The symmetry around 0% indicates balanced performance deviations, though the slight asymmetry in d=16 warrants exploration. The declining peak frequency with increasing 'd' might reflect diminishing class diversity or stricter performance constraints at higher 'd' values. </details> Figure 12: Progression of relative per-class accuracy vs ${\rm MRL}$ -2048. As the dimensionality increases, the spread shrinks while the class marked (x) (Madagascar cat) loses accuracy. In an attempt to derive an optimal routing policy to emulate oracle accuracy, we designed the adaptive classification via cascading method as discussed in Appendix D.1. This led to an interesting observation on the expected dimensionality for $76.30\$ top-1 classification accuracy being just $d\sim 37$ . We leave the design and learning of a more optimal policy for future work. Table 22: Percentage of ImageNet-1K validation set that is first correctly predicted using each representation size $d$ . We note that $18.46\$ of the samples cannot be correctly predicted by any representation size. The remaining $81.54\$ constitutes the oracle accuracy. | Correctly Predicted | 67.46 | 8.78 | 2.58 | 1.35 | 0.64 | 0.31 | 0.20 | 0.12 | 0.06 | 18.46 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | Table 23: Oracle classification accuracy of various evaluation datasets for ResNet50– ${\rm MRL}$ model trained on ImageNet-1K. | FF–2048 ${\rm MRL}$ –Oracle | 76.9 81.5 | 64.9 70.6 | 3.6 8.7 | 35.1 39.8 | 23.7 28.9 | | --- | --- | --- | --- | --- | --- | Grad-CAM Examples We analyzed the nature of model disagreement across representation sizes with ${\rm MRL}$ models with the help of Grad-CAM visualization [80]. We observed there were certain classes in ImageNet-1K such as "tools", "vegetables" and "meat cutting knife" which were occasionally located around multiple objects and a cluttered environment. In such scenarios, we observed that smaller representation size models would often get confused due to other objects and fail to extract the object of interest which generated the correct label. We also observed a different nature of disagreement arising when the models got confused within the same superclass. For example, ImageNet-1K has multiple "snake" classes, and models often confuse a snake image for an incorrect species of snake. Superclass Performance We created a 30 superclass subset of the validation set based on wordnet hierarchy (Table 24) to quantify the performance of ${\rm MRL}$ model on ImageNet-1K superclasses. Table 25 quantifies the performance with different representation size. Table 24: 30 Superclasses in ImageNet-1K corresponding to the performance in Table 25. | insect | motor vehicle | artiodactyl | vegetable | game equipment | | --- | --- | --- | --- | --- | | terrier | serpent | machine | measuring device | sheepdog | | protective covering | sporting dog | vessel, watercraft | building | lizard | | garment | hound | monkey | home appliance | wind instrument | | vessel | fish | nourishment | electronic equipment | oscine | | furniture | wading bird | tool | canine | mechanism | Table 25: Performance of ${\rm MRL}$ model on 31-way classification (1 extra class is for reject token) on ImageNet-1K superclasses. | ${\rm MRL}$ | 85.57 | 88.67 | 89.48 | 89.82 | 89.97 | 90.11 | 90.18 | 90.22 | 90.21 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ## Appendix K Ablation Studies ### K.1 ${\rm MRL}$ Training Paradigm Table 26: Top-1 classification accuracy (%) on ImageNet-1K of various ResNet50 models which are finetuned on pretrained FF-2048 model. We observed that adding more non-linearities is able to induce nesting to a reasonable extent even if the model was not pretrained with nesting in mind. | 8 16 32 | 5.15 13.79 32.52 | 36.11 58.42 67.81 | 54.78 67.26 71.62 | 60.02 70.10 72.84 | 66.63 73.53 75.03 | | --- | --- | --- | --- | --- | --- | | 64 | 52.66 | 72.42 | 73.61 | 74.29 | 75.82 | | 128 | 64.60 | 74.41 | 74.67 | 75.03 | 76.30 | | 256 | 69.29 | 75.30 | 75.23 | 75.38 | 76.47 | | 512 | 70.51 | 75.96 | 75.47 | 75.64 | 76.65 | | 1024 | 70.19 | 76.18 | 75.70 | 75.75 | 76.76 | | 2048 | 69.72 | 76.44 | 75.96 | 75.97 | 76.80 | ${\rm Matryoshka~Representations}$ via Finetuning. To observe if nesting can be induced in models that were not explicitly trained with nesting from scratch, we loaded a pretrained FF-2048 ResNet50 model and initialized a new ${\rm MRL}$ layer, as defined in Algorithm 2, Appendix C. We then unfroze different layers of the backbone to observe how much non-linearity in the form of unfrozen conv layers needed to be present to enforce nesting into a pretrained FF model. A description of these layers can be found in the ResNet50 architecture [29]. All models were finetuned with the FFCV pipeline, with same training configuration as in the end-to-end training aside from changing lr $=0.1$ and epochs $=10$ . We observed that finetuning the linear layer alone was insufficient to learn ${\rm Matryoshka~Representations}$ at lower dimensionalities. Adding more and more non-linear conv+ReLU layers steadily improved classification accuracy of $d=8$ from $5\$ to $60\$ after finetuning, which was only $6\$ less than training ${\rm MRL}$ end-to-end for 40 epochs. This difference was successively less pronounced as we increased dimensionality past $d=64$ , to within $1.5\$ for all larger dimensionalities. The full results of this ablation can be seen in Table 26. Table 27: An ablation over boosting training loss at lower nesting dimensions, with top-1 and top-5 accuracy (%). The models are described in Appendix K.1. | 8 16 32 | 66.63 73.53 75.03 | 84.66 89.52 91.31 | 69.53 73.86 75.28 | 86.19 89.44 91.21 | 69.24 73.91 75.10 | 85.96 89.55 91.14 | | --- | --- | --- | --- | --- | --- | --- | | 64 | 75.82 | 92.27 | 75.84 | 92.22 | 75.67 | 92.06 | | 128 | 76.30 | 92.82 | 76.28 | 92.74 | 76.07 | 92.52 | | 256 | 76.47 | 93.02 | 76.48 | 92.97 | 76.22 | 92.72 | | 512 | 76.65 | 93.13 | 76.56 | 93.09 | 76.35 | 92.85 | | 1024 | 76.76 | 93.22 | 76.71 | 93.21 | 76.39 | 92.98 | | 2048 | 76.80 | 93.32 | 76.76 | 93.28 | 76.52 | 93.05 | Relative Importance. We performed an ablation of ${\rm MRL}$ over the relative importance, $c_{m}$ , of different nesting dimensions $m\in\cal{M}$ , as defined in Sec. 3. In an attempt to improve performance at lower dimensionalities, we boosted the relative importance $c_{m}$ of training loss at lower dimensions as in Eq. 1 with two models, ${\rm MRL}$ -8boost and ${\rm MRL}$ -8+16boost. The ${\rm MRL}$ -8boost model had $c_{m\in\cal M}=[2,1,1,1,1,1,1,1,1]$ and the ${\rm MRL}$ -8+16boost model had $c_{m\in\cal M}=[2,1.5,1,1,1,1,1,1,1]$ . The relative importance list $c_{m\in\cal M}$ had a 1-to-1 correspondence with nesting dimension set $\mathcal{M}$ . In Table 27, we observed that ${\rm MRL}$ -8boost improves top-1 accuracy by $3\$ at $d=8$ , and also improves top-1 accuracy of all representation scales from 16 to 256 over ${\rm MRL}$ , while only hurting the performance at 512 to 2048 representation scales by a maximum of 0.1%. This suggests that the relative importance $c_{m}$ can be tuned/set for optimal accuracy for all $m\in\mathcal{M}$ , but we leave this extension for future work. ${\rm Matryoshka~Representations}$ at Arbitrary Granularities. To train ${\rm MRL}$ , we used nested dimensions at logarithmic granularities $\mathcal{M}=\{8,16,\ldots,1024,2048\}$ as detailed in Section 3. We made this choice for two empirically-driven reasons: a) The accuracy improvement with increasing representation size was more logarithmic than linear (as shown by FF models in Figure 3). This indicated that optimizing for granularities increasing in a non-logarithmic fashion would be sub-optimal both for maximum performance and expected efficiency; b) If we have $m$ arbitrary granularities, the expected cost of the linear classifier to train ${\rm MRL}$ scales as $O(L*(m^{2}))$ while logarithmic granularities result in $O(L*2log(d))$ space and compute costs. To demonstrate this effect, we learned ${\rm Matryoshka~Representations}$ with uniform ( ${\rm MRL}$ -Uniform) nesting dimensions $m\in\mathcal{M}=\{8,212,416,620,824,1028,1232,1436,1640,1844,2048\}$ . We evaluated this model at the standard ( ${\rm MRL}$ -log) dimensions $m\in\mathcal{M}=\{8,16,32,64,128,256,512,1024,2048\}$ for ease of comparison to reported numbers using 1-NN accuracy (%). As shown in Table 29, we observed that while performance interpolated, ${\rm MRL}$ -Uniform suffered at low dimensions as the logarithmic spacing of ${\rm MRL}$ -log resulted in tighter packing of information in these initial dimensions. The higher nesting dimensions of ${\rm MRL}$ -Uniform did not help in significant accuracy improvement due to accuracy saturation, which is often logarithmic in representation size as shown by FF models. Note that the slight improvement at dimensions higher than 512 for ${\rm MRL}$ -Uniform is due to multiple granularities around them compared to just three for ${\rm MRL}$ -log, which are not useful in practice for efficiency. Lower Dimensionality. We experimented with training ${\rm MRL}$ with smaller nesting dimension than $m=8$ , as shown in Table 29, with two models: MRL-4 and MRL-6. We found that using lower than 8-dimensions to train ${\rm MRL}$ , i.e. $m_{0}\in\{4,6\}$ for MRL-4 and MRL-6 respectively, did not affect the top-1 accuracy of other granularities significantly. However, granularities smaller than 8-dimensions had very low accuracy and were often unusable for deployment along with additional training difficulty. We also observed a small dip in accuracy at higher dimensions which we attribute to the joint loss that now also included the harder optimization of the smallest dimension. Lastly, we hypothesize the dimensionality of 8 is an empirically validated design choice due to the considerable accuracy it provided along with the ease of training. Table 28: An ablation over training with smaller nesting dimensionalities in terms of Top-1 accuracy (%). MRL-4 and MRL-6 are variations of the original model (MRL-8) with $m_{0}\in\{4,6\}$ , where $m\in\mathcal{M}$ is part of the nesting_list as seen in Alg 2. | 4 6 8 | 27.25 - 66.86 | - 58.71 67.55 | - - 66.63 | | --- | --- | --- | --- | | 16 | 73.36 | 73.10 | 73.53 | | 32 | 74.82 | 74.49 | 75.03 | | 64 | 75.51 | 75.32 | 75.82 | | 128 | 75.93 | 75.61 | 76.30 | | 256 | 76.08 | 75.82 | 76.47 | | 512 | 76.31 | 75.93 | 76.65 | | 1024 | 76.38 | 76.04 | 76.76 | | 2048 | 76.43 | 76.12 | 76.80 | Table 29: An ablation over training ${\rm MRL}$ with nesting list at uniformly distributed granularities. Entries in the ${\rm MRL}$ -Uniform column are evaluated at logarithmic dimensions for a fair comparison to ${\rm MRL}$ -Log (standard ${\rm MRL}$ ) with 1-NN accuracy (%). | 8 16 32 | 62.19 67.91 69.46 | 58.44 61.11 63.82 | | --- | --- | --- | | 64 | 70.17 | 66.44 | | 128 | 70.52 | 68.71 | | 256 | 70.62 | 70.06 | | 512 | 70.82 | 70.98 | | 1024 | 70.89 | 71.37 | | 2048 | 70.97 | 71.44 | ### K.2 Retrieval Adaptive Retrieval. To examine the effect of increasing shortlist lengths on search time, we performed a reranking ablation over shortlist lengths for $D_{s}$ = 16 and $D_{r}$ = 2048 over ImageNet-1K in Table 30, and over ImageNet-4K in Table 31. We observed that using a larger shortlist $k$ saturated ImageNet-1K performance at $k$ =200. But using larger shortlists until $k=2048$ , the maximum value supported by the FAISS framework, steadily improved performance on ImageNet-4K. This is likely due to the increased database size, but could also indicate a correlation with ImageNet-4K being slightly out-of-distribution making the task at hand harder. Table 30: Adaptive retrieval ablation over shortlist length $k$ for $D_{s}=16$ , $D_{r}=2048$ on ImageNet-1K with exact search. Entries with the highest P@1 and mAP@10 across all $k$ are in bold. | 100 200 400 | 70.88 70.90 70.94 | 65.19 65.27 65.26 | 63.62 63.73 63.71 | 62.59 62.82 62.81 | 61.24 61.97 62.03 | 69.96 70.10 70.15 | 69.24 69.44 69.51 | 68.53 68.90 69.02 | 67.20 68.21 68.47 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 800 | 70.96 | 65.23 | 63.64 | 62.69 | 61.85 | 70.16 | 69.52 | 69.02 | 68.45 | | 1600 | 70.96 | 65.20 | 63.58 | 62.58 | 61.66 | 70.16 | 69.5 | 68.97 | 68.36 | | 2048 | 70.97 | 65.20 | 63.57 | 62.58 | 61.64 | 70.16 | 69.5 | 68.97 | 68.35 | Table 31: Adaptive retrieval ablation over shortlist length $k$ for $D_{s}=16$ , $D_{r}=2048$ on ImageNet-4K with exact search. | 100 200 400 | 27.70 28.56 29.34 | 14.38 15.21 15.83 | 10.62 11.43 12.06 | 8.26 9.11 9.76 | 6.07 7.12 7.79 | 20.12 21.23 22.08 | 16.87 18.13 19.09 | 14.29 15.73 16.83 | 11.26 13.27 14.54 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 800 | 29.86 | 16.30 | 12.53 | 10.23 | 8.26 | 22.72 | 19.83 | 17.65 | 15.45 | | 1600 | 30.24 | 16.63 | 12.86 | 10.56 | 8.60 | 23.18 | 20.36 | 18.23 | 16.11 | | 2048 | 30.35 | 16.73 | 12.96 | 10.65 | 8.69 | 23.31 | 20.50 | 18.40 | 16.30 |