2205.13147v4

# 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 $\mathbf{14}×$ smaller embedding size for ImageNet-1K classification at the same level of accuracy; (b) up to $\mathbf{14}×$ 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 Training ### Overview The image is a diagram illustrating a process involving inference and training, likely within a machine learning context. It shows the flow of data and operations between "Inference" and "Training" stages, highlighting adaptive retrieval and classification. ### Components/Axes * **Title:** The diagram is implicitly titled "Adaptive Retrieval and Classification Training" based on the elements depicted. * **Sections:** The diagram is divided into two main sections: "Inference" (left) and "Training" (right). * **Inference Block:** Contains "Shortlisting" and "Re-ranking" steps within an "Adaptive Retrieval" box. Also contains an "Adaptive Classification" box. * **Training Block:** Features a vertical stack of colored rectangles representing different levels or components of a variable 'z', where z ∈ R^d. These components are associated with loss functions L(z1:d/16), L(z1:d/8), L(z1:d/4), L(z1:d/2), and L(z1:d). * **Connections:** Arrows indicate the flow of information between the blocks. * **Loss Function Summation:** A summation symbol (⊕) combines the loss functions to produce a final loss L(z). ### Detailed Analysis * **Inference - Adaptive Retrieval:** * "Shortlisting" (top, light blue box) feeds into "Re-ranking" (green box). * Both are contained within a rounded-corner box labeled "Adaptive Retrieval" (tan background). * **Inference - Adaptive Classification:** * Located below "Adaptive Retrieval" (tan background). * Contains a series of stacked rectangles, similar to the "Training" block, but arranged in a pyramid-like structure. The rectangles are colored red, orange, blue, and yellow. * A dashed line connects the top of the pyramid to the bottom, suggesting a progressive classification process. * **Training - Variable z:** * A vertical stack of colored rectangles, enclosed in a grey border. From top to bottom, the rectangles are red, orange, blue, and yellow. A small red/white icon is at the bottom. * Each rectangle corresponds to a loss function: * Red: L(z1:d/16) * Orange: L(z1:d/8) * Blue: L(z1:d/4) * Yellow: L(z1:d/2) * Grey: L(z1:d) * **Connections:** * An orange arrow connects "Shortlisting" to the red rectangle in the "Training" block. * A grey arrow connects "Re-ranking" to the orange rectangle in the "Training" block. * A black arrow connects "Adaptive Classification" to the grey rectangle in the "Training" block. * A blue arrow connects the blue rectangle to the summation symbol. * A yellow arrow connects the yellow rectangle to the summation symbol. * The summation symbol (⊕) combines the loss functions and outputs L(z). ### Key Observations * The diagram illustrates a multi-stage process where inference results are used to train a model. * The "Training" block represents a hierarchical structure of the variable 'z', with different levels contributing to the overall loss function. * The "Adaptive Classification" block mirrors the structure in the "Training" block, suggesting a relationship between classification granularity and the variable's components. ### Interpretation The diagram depicts a system where inference (adaptive retrieval and classification) informs the training of a model. The model's training process involves a variable 'z' that is decomposed into different levels (z1:d/16, z1:d/8, z1:d/4, z1:d/2, z1:d), each associated with a loss function. The adaptive retrieval process, through shortlisting and re-ranking, likely selects relevant components of 'z' for training. The adaptive classification process provides further information to refine the training. The summation of individual loss functions suggests a joint optimization strategy. The diagram highlights the adaptive nature of both the retrieval and classification processes, indicating that the system dynamically adjusts its behavior based on the input data. </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, $m∈[d]$ , 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 $14×$ 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 $128×$ theoretical (in terms of FLOPS) and $14×$ 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 $14×$ 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 $d∈\mathbb{N}$ , consider a set $\mathcal{M}⊂[d]$ of representation sizes. For a datapoint $x$ in the input domain $\mathcal{X}$ , our goal is to learn a $d$ -dimensional representation vector $z∈\mathbb{R}^{d}$ . For every $m∈\mathcal{M}$ , ${\rm Matryoshka~Representation~Learning}$ ( ${\rm MRL}$ ) enables each of the first $m$ dimensions of the embedding vector, $z_{1:m}∈\mathbb{R}^{m}$ to be independently capable of being a transferable and general purpose representation of the datapoint $x$ . We obtain $z$ using a deep neural network $F(\,·\,;\theta_{F})\colon\mathcal{X}→\mathbb{R}^{d}$ 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., $\lvert\mathcal{M}\rvert≤\left\lfloor\log(d)\right\rfloor$ . 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 $224× 224$ 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 $\mathcal{M}=\{8,16,...,1024,2048\}$ as the nesting dimensions. Suppose we are given a labelled dataset $\mathcal{D}=\{(x_{1},y_{1}),...,(x_{N},y_{N})\}$ where $x_{i}∈\mathcal{X}$ is an input point and $y_{i}∈[L]$ is the label of $x_{i}$ for all $i∈[N]$ . ${\rm MRL}$ optimizes the multi-class classification loss for each of the nested dimension $m∈\mathcal{M}$ using standard empirical risk minimization using a separate linear classifier, parameterized by $\mathbf{W}^{(m)}∈\mathbb{R}^{L× m}$ . All the losses are aggregated after scaling with their relative importance $\left(c_{m}≥ 0\right)_{m∈\mathcal{M}}$ 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 ${\cal L}\colon\mathbb{R}^{L}×[L]→\mathbb{R}_{+}$ 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 $m∈\mathcal{M}$ ; 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 $\mathbf{W}∈\mathbb{R}^{L× d}$ . 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 Chart: Top-1 Accuracy vs. Representation Size ### Overview The image is a line chart comparing the Top-1 Accuracy (%) of different models (MRL, MRL-E, FF, SVD, Slim. Net, and Rand. LP) against varying Representation Sizes. The x-axis represents the Representation Size, while the y-axis represents the Top-1 Accuracy in percentage. ### Components/Axes * **X-axis:** Representation Size, with values 8, 16, 32, 64, 128, 256, 512, 1024, and 2048. * **Y-axis:** Top-1 Accuracy (%), with values ranging from 40 to 80, in increments of 10. * **Legend:** Located on the right side of the chart, it identifies each model with a specific color and marker: * MRL (Blue, Circle marker) * MRL-E (Orange, Triangle marker) * FF (Green, Triangle marker pointing to the right) * SVD (Red, Circle marker) * Slim. Net (Purple, Plus marker) * Rand. LP (Brown, X marker) ### Detailed Analysis * **MRL (Blue, Circle marker):** The line starts at approximately 66% accuracy at a representation size of 8, increases to about 74% at 16, reaches around 76% at 32, and plateaus around 77% for representation sizes 64 and above. * **MRL-E (Orange, Triangle marker):** The line starts at approximately 56% accuracy at a representation size of 8, increases to about 72% at 16, reaches around 75% at 32, and plateaus around 76% for representation sizes 64 and above. * **FF (Green, Triangle marker pointing to the right):** The line starts at approximately 66% accuracy at a representation size of 8, increases to about 74% at 16, reaches around 76% at 32, and plateaus around 77% for representation sizes 64 and above. * **SVD (Red, Circle marker):** The line starts at approximately 48% accuracy at a representation size of 64, increases to about 67% at 128, reaches around 73% at 256, and plateaus around 76% for representation sizes 512 and above. * **Slim. Net (Purple, Plus marker):** The line starts at approximately 40% accuracy at a representation size of 256, increases to about 75% at 1024, and plateaus around 76% for representation sizes 2048. * **Rand. LP (Brown, X marker):** The line starts at approximately 40% accuracy at a representation size of 64, increases to about 72% at 256, and plateaus around 76% for representation sizes 512 and above. ### Key Observations * MRL, MRL-E, and FF models achieve relatively high accuracy even at smaller representation sizes (8-32). * SVD and Rand. LP models show a significant increase in accuracy as the representation size increases from 64 to 256. * Slim. Net model shows a significant increase in accuracy as the representation size increases from 256 to 1024. * All models tend to plateau in accuracy as the representation size increases beyond 256. ### Interpretation The chart illustrates the relationship between representation size and Top-1 accuracy for different models. It suggests that increasing the representation size generally improves accuracy, but the improvement diminishes beyond a certain point. The models MRL, MRL-E, and FF appear to be more effective at smaller representation sizes compared to SVD, Slim. Net, and Rand. LP. The plateauing effect indicates that there is a limit to how much accuracy can be gained by simply increasing the representation size, and other factors may become more important for further improvement. </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 Chart: 1-NN Accuracy vs. Representation Size ### Overview The image is a line chart comparing the 1-Nearest Neighbor (1-NN) accuracy of different methods (MRL, MRL-E, FF, SVD, Slim. Net, and Rand. FS) across varying representation sizes. The x-axis represents the representation size, while the y-axis represents the 1-NN accuracy in percentage. ### Components/Axes * **Title:** There is no explicit title on the chart. * **X-axis:** * Label: "Representation Size" * Scale: 8, 16, 32, 64, 128, 256, 512, 1024, 2048 * **Y-axis:** * Label: "1-NN Accuracy (%)" * Scale: 40, 50, 60, 70 * **Legend:** Located on the right side of the chart, listing the methods and their corresponding line styles and colors: * MRL (Blue line with circle markers) * MRL-E (Orange dashed line with triangle markers) * FF (Green dash-dot line with inverted triangle markers) * SVD (Red dotted line with circle markers) * Slim. Net (Purple dashed line with plus markers) * Rand. FS (Brown solid line with x markers) ### Detailed Analysis * **MRL (Blue line with circle markers):** The line starts at approximately 62% accuracy at a representation size of 8, increases to about 67% at 16, reaches approximately 69% at 32, and then plateaus around 71-72% for larger representation sizes. * (8, 62%) * (16, 67%) * (32, 69%) * (64, 70%) * (128, 71%) * (256, 71%) * (512, 71%) * (1024, 72%) * (2048, 72%) * **MRL-E (Orange dashed line with triangle markers):** The line starts at approximately 58% accuracy at a representation size of 8, increases to about 66% at 16, reaches approximately 69% at 32, and then plateaus around 70-71% for larger representation sizes. * (8, 58%) * (16, 66%) * (32, 69%) * (64, 70%) * (128, 70%) * (256, 71%) * (512, 71%) * (1024, 71%) * (2048, 71%) * **FF (Green dash-dot line with inverted triangle markers):** The line starts at approximately 59% accuracy at a representation size of 8, increases to about 67% at 16, reaches approximately 69% at 32, and then plateaus around 70-71% for larger representation sizes. * (8, 59%) * (16, 67%) * (32, 69%) * (64, 70%) * (128, 70%) * (256, 70%) * (512, 71%) * (1024, 71%) * (2048, 71%) * **SVD (Red dotted line with circle markers):** The line starts at approximately 42% accuracy at a representation size of 8, increases to about 47% at 16, reaches approximately 60% at 32, and then plateaus around 69-70% for larger representation sizes. * (8, 42%) * (16, 47%) * (32, 60%) * (64, 67%) * (128, 69%) * (256, 70%) * (512, 70%) * (1024, 70%) * (2048, 71%) * **Slim. Net (Purple dashed line with plus markers):** The line starts at approximately 40% accuracy at a representation size of 64, increases to about 58% at 128, reaches approximately 64% at 256, and then plateaus around 65-66% for larger representation sizes. * (64, 40%) * (128, 58%) * (256, 64%) * (512, 65%) * (1024, 65%) * (2048, 66%) * **Rand. FS (Brown solid line with x markers):** The line starts at approximately 40% accuracy at a representation size of 64, increases to about 61% at 128, reaches approximately 67% at 256, and then plateaus around 70-71% for larger representation sizes. * (64, 40%) * (128, 61%) * (256, 67%) * (512, 70%) * (1024, 71%) * (2048, 72%) ### Key Observations * MRL, MRL-E, and FF methods achieve relatively high accuracy even with small representation sizes. * SVD, Slim. Net, and Rand. FS methods show a more gradual increase in accuracy as the representation size increases. * All methods tend to converge to a similar accuracy range (70-72%) as the representation size becomes larger. * Slim. Net consistently underperforms compared to the other methods. ### Interpretation The chart demonstrates the relationship between representation size and 1-NN accuracy for different methods. MRL, MRL-E, and FF appear to be more efficient in utilizing smaller representation sizes to achieve high accuracy, while SVD, Slim. Net, and Rand. FS require larger representation sizes to reach comparable performance. The convergence of all methods at larger representation sizes suggests that there may be a limit to the accuracy achievable with these methods, regardless of the representation size. The consistently lower performance of Slim. Net indicates that it may be less effective than the other methods in this context. </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 Chart: 1-NN Accuracy vs. Representation Size ### Overview The image is a line chart comparing the 1-Nearest Neighbor (1-NN) accuracy of different models (JFT and ALIGN) using various representation learning techniques (MRL, MRL-E, and Rand.) across different representation sizes. The x-axis represents the representation size, and the y-axis represents the 1-NN accuracy in percentage. ### Components/Axes * **X-axis:** Representation Size, with values 12, 24, 48, 96, 192, 384, and 768. * **Y-axis:** 1-NN Accuracy (%), with a scale from 20 to 80 in increments of 20. * **Legend (Top-right):** * Blue line with circles: JFT MRL * Orange dashed line with triangles: ALIGN MRL * Green dashed line with inverted triangles: JFT MRL-E * Red dotted line with circles: JFT Rand. * Purple dashed-dotted line with plus signs: ALIGN Rand. ### Detailed Analysis * **JFT MRL (Blue):** Starts at approximately 53% accuracy at representation size 12, increases to around 63% at 24, 68% at 48, 70% at 96, and plateaus around 72-73% for representation sizes 192, 384, and 768. * **ALIGN MRL (Orange):** Starts at approximately 43% accuracy at representation size 12, increases to around 58% at 24, 65% at 48, 67% at 96, and plateaus around 68-70% for representation sizes 192, 384, and 768. * **JFT MRL-E (Green):** Starts at approximately 53% accuracy at representation size 12, increases to around 63% at 24, 68% at 48, 70% at 96, and plateaus around 72-73% for representation sizes 192, 384, and 768. * **JFT Rand. (Red):** Starts at approximately 28% accuracy at representation size 12, increases to around 47% at 24, 61% at 48, 67% at 96, and plateaus around 70-71% for representation sizes 192, 384, and 768. * **ALIGN Rand. (Purple):** Starts at approximately 12% accuracy at representation size 12, increases to around 35% at 24, 50% at 48, 60% at 96, and plateaus around 65-67% for representation sizes 192, 384, and 768. ### Key Observations * JFT MRL and JFT MRL-E perform similarly and consistently achieve the highest 1-NN accuracy across all representation sizes. * ALIGN MRL performs slightly worse than JFT MRL and JFT MRL-E. * JFT Rand. performs better than ALIGN Rand. * All models show a significant increase in accuracy as the representation size increases from 12 to 96, after which the accuracy plateaus. * The "Rand." methods (JFT Rand. and ALIGN Rand.) start with the lowest accuracy but eventually converge towards the performance of the MRL methods as the representation size increases. ### Interpretation The chart demonstrates the impact of representation size on the 1-NN accuracy of different models and representation learning techniques. The MRL and MRL-E techniques consistently outperform the random initialization ("Rand.") methods, especially at smaller representation sizes. As the representation size increases, the performance gap between the MRL and Rand. methods narrows, suggesting that larger representation sizes can compensate for less effective initialization strategies. The plateauing of accuracy beyond a representation size of 96 indicates diminishing returns, suggesting that further increases in representation size may not significantly improve performance. The similarity in performance between JFT MRL and JFT MRL-E suggests that the "E" variant does not offer a substantial advantage in terms of 1-NN accuracy. </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 Chart: 1-NN Accuracy vs. Representation Size ### Overview The image is a line chart comparing the 1-Nearest Neighbor (1-NN) accuracy of different models (ViT-ALIGN, ViT-JFT, RN50-IN1K) and their interpolated versions (Int) across varying representation sizes. The x-axis represents the representation size, and the y-axis represents the 1-NN accuracy in percentage. ### Components/Axes * **X-axis:** Representation Size, with values 8, 16, 32, 64, 128, 256, 512, 1024, 2048. * **Y-axis:** 1-NN Accuracy (%), with a scale from 50% to 70%. * **Legend:** Located on the right side of the chart, identifying each line by model type and interpolation status: * Green, dash-dot line with triangle markers: ViT-ALIGN * Blue, dashed line with circle markers: ViT-JFT * Purple, dashed line with square markers: RN50-IN1K * Red, dash-dot line with inverted triangle markers: ViT-ALIGN-Int * Pink/Red, dashed line with circle markers: ViT-JFT-Int * Red, dashed line with square markers: RN50-IN1K-Int ### Detailed Analysis * **ViT-ALIGN (Green, dash-dot line with triangle markers):** The accuracy starts at approximately 44% at representation size 8, increases sharply to about 57% at 32, then gradually rises to approximately 68% at 1024, and plateaus around 68% at 2048. * **ViT-JFT (Blue, dashed line with circle markers):** The accuracy starts at approximately 53% at representation size 8, increases to about 63% at 32, then gradually rises to approximately 71% at 1024, and plateaus around 71% at 2048. * **RN50-IN1K (Purple, dashed line with square markers):** The accuracy starts at approximately 62% at representation size 8, increases to about 69% at 32, then gradually rises to approximately 71% at 1024, and plateaus around 71% at 2048. * **ViT-ALIGN-Int (Red, dash-dot line with inverted triangle markers):** The accuracy starts at approximately 49% at representation size 16, increases sharply to about 64% at 64, then gradually rises to approximately 66% at 256, and plateaus around 66% at 2048. * **ViT-JFT-Int (Pink/Red, dashed line with circle markers):** The accuracy starts at approximately 58% at representation size 16, increases to about 66% at 64, then gradually rises to approximately 72% at 512, and plateaus around 72% at 2048. * **RN50-IN1K-Int (Red, dashed line with square markers):** The accuracy starts at approximately 65% at representation size 16, increases to about 70% at 32, then gradually rises to approximately 72% at 256, and plateaus around 72% at 2048. ### Key Observations * All models show an increase in 1-NN accuracy as the representation size increases. * The rate of increase in accuracy diminishes as the representation size grows larger, indicating diminishing returns. * RN50-IN1K and ViT-JFT models generally outperform ViT-ALIGN at larger representation sizes. * The interpolated versions (Int) of the models generally have lower accuracy at smaller representation sizes but converge to similar or slightly higher accuracy at larger representation sizes compared to their non-interpolated counterparts. ### Interpretation The chart illustrates the trade-off between representation size and 1-NN accuracy for different models. Increasing the representation size generally improves accuracy, but the improvement plateaus beyond a certain point. The choice of model and representation size should consider the balance between accuracy and computational cost. The interpolated versions of the models suggest that interpolation techniques can be effective in improving accuracy, especially at larger representation sizes. The RN50-IN1K and ViT-JFT models appear to be more effective in achieving higher accuracy compared to ViT-ALIGN, based on this data. </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 $8→ 16→ 32$ ) 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 $\sim 14×$ 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 ## Chart: Top-1 Accuracy vs. Representation Size ### Overview The image is a chart comparing the Top-1 Accuracy (%) of different models (MRL-AC and FF) against the (Expected) Representation Size. The chart also includes a horizontal line representing the performance of FF 2048. An annotation indicates a "14x smaller representation size". ### Components/Axes * **Y-axis:** Top-1 Accuracy (%), ranging from 74% to 77%. * **X-axis:** (Expected) Representation Size, with values 16, 32, 64, 128, 256, and 512. * **Legend:** Located in the bottom-right corner, it identifies the data series: * Blue circles: MRL-AC * Orange crosses: FF * Purple dash-dotted line: FF 2048 * **Annotation:** "14x smaller representation size" with a green dashed line and arrow pointing from the MRL-AC data point at representation size 32 to the MRL-AC data point at representation size 512. ### Detailed Analysis * **MRL-AC (Blue Circles):** The Top-1 Accuracy generally increases as the Representation Size increases, but plateaus after a representation size of 32. * At 16: ~75.2% * At 32: ~76.1% * At 64: ~76.4% * At 128: ~76.4% * At 256: ~76.4% * At 512: ~76.4% * **FF (Orange Crosses):** The Top-1 Accuracy increases as the Representation Size increases. * At 32: ~74.7% * At 64: ~75.4% * At 128: ~75.5% * At 256: ~75.7% * At 512: ~76.4% * **FF 2048 (Purple Dash-Dotted Line):** This line is horizontal, indicating a constant Top-1 Accuracy regardless of the X-axis. * Accuracy: ~77.1% ### Key Observations * MRL-AC achieves a relatively high accuracy with smaller representation sizes compared to FF. * FF 2048 has the highest accuracy overall. * The annotation highlights that MRL-AC can achieve similar performance to FF with a significantly smaller representation size (14x smaller). ### Interpretation The chart demonstrates the trade-off between model accuracy and representation size. MRL-AC appears to be more efficient in terms of representation size, achieving comparable accuracy to FF with smaller sizes. However, FF 2048, likely a larger model, achieves the highest accuracy. The "14x smaller representation size" annotation suggests that MRL-AC can achieve a similar accuracy to FF with a much smaller model size, which could be beneficial in resource-constrained environments. The plateauing of MRL-AC's accuracy after a representation size of 32 suggests that increasing the representation size beyond this point does not significantly improve performance. </details> Figure 6: Adaptive classification on ${\rm MRL}$ ResNet50 using cascades results in $14×$ 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 Chart: mAP@10 vs. Representation Size ### Overview The image is a line chart comparing the performance of different methods (MRL, MRL-E, FF, SVD, Slim. Net, and Rand. FS) based on their mean Average Precision at 10 (mAP@10) as the representation size increases. The x-axis represents the representation size, and the y-axis represents the mAP@10 in percentage. ### Components/Axes * **Title:** There is no explicit title on the chart. * **X-axis:** Representation Size. The x-axis is labeled "Representation Size" and has the following tick marks: 8, 16, 32, 64, 128, 256, 512, 1024, 2048. * **Y-axis:** mAP@10 (%). The y-axis is labeled "mAP@10 (%)" and has tick marks at 40, 45, 50, 55, 60, and 65. * **Legend:** Located on the right side of the chart. * **MRL:** Solid blue line with circle markers. * **MRL-E:** Dashed orange line with triangle markers. * **FF:** Dashed green line with inverted triangle markers. * **SVD:** Dotted red line with circle markers. * **Slim. Net:** Dashed-dotted purple line with plus markers. * **Rand. FS:** Solid brown line with cross markers. ### Detailed Analysis * **MRL (Solid Blue Line):** * Trend: Increases sharply from 8 to 16, then plateaus. * Data Points: * 8: ~57% * 16: ~63% * 32: ~64% * 64: ~65% * 128: ~65% * 256: ~65% * 512: ~65% * 1024: ~65% * 2048: ~64% * **MRL-E (Dashed Orange Line):** * Trend: Increases sharply from 8 to 32, then plateaus with a slight decrease at the end. * Data Points: * 8: ~52% * 16: ~62% * 32: ~64% * 64: ~64% * 128: ~63% * 256: ~64% * 512: ~64% * 1024: ~63% * 2048: ~63% * **FF (Dashed Green Line):** * Trend: Increases sharply from 8 to 32, then plateaus with a slight decrease at the end. * Data Points: * 8: ~54% * 16: ~62% * 32: ~64% * 64: ~64% * 128: ~63% * 256: ~62% * 512: ~62% * 1024: ~61% * 2048: ~63% * **SVD (Dotted Red Line):** * Trend: Increases steadily from 8 to 1024. * Data Points: * 8: ~40% * 16: ~40% * 32: ~50% * 64: ~58% * 128: ~62% * 256: ~63% * 512: ~64% * 1024: ~63% * 2048: ~64% * **Slim. Net (Dashed-Dotted Purple Line):** * Trend: Increases from 8 to 2048. * Data Points: * 8: ~40% * 16: ~40% * 32: ~40% * 64: ~40% * 128: ~40% * 256: ~42% * 512: ~50% * 1024: ~54% * 2048: ~55% * **Rand. FS (Solid Brown Line):** * Trend: Increases from 8 to 2048. * Data Points: * 8: ~40% * 16: ~40% * 32: ~40% * 64: ~40% * 128: ~50% * 256: ~53% * 512: ~58% * 1024: ~61% * 2048: ~64% ### Key Observations * MRL, MRL-E, and FF achieve high mAP@10 with smaller representation sizes and plateau quickly. * SVD, Slim. Net, and Rand. FS require larger representation sizes to achieve comparable performance. * MRL consistently performs well across all representation sizes. * Slim. Net and Rand. FS start with very low mAP@10 values and gradually improve with increasing representation size. ### Interpretation The chart illustrates the trade-off between representation size and performance (mAP@10) for different methods. MRL, MRL-E, and FF are more efficient in terms of representation size, achieving high performance with smaller representations. SVD, Slim. Net, and Rand. FS require larger representation sizes to reach similar performance levels. This suggests that MRL, MRL-E, and FF may be more suitable for applications where representation size is a constraint. The performance of Slim. Net and Rand. FS is significantly lower at smaller representation sizes, indicating that they may require more data or more complex models to achieve good performance. </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 $≤ 256$ 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 ### Overview The image is a scatter plot showing the relationship between mAP@10 (mean Average Precision at 10) and MFLOPS/Query (Millions of Floating Point Operations per Query). The plot includes data points represented by circles of varying sizes and colors, a dashed line representing the "Funnel" data series, and annotations indicating theoretical and real-world speed-ups. ### Components/Axes * **X-axis:** MFLOPS/Query. The scale is logarithmic, with marked values at 10² and 10³. * **Y-axis:** mAP@10 (%). The scale ranges from 64.9 to 65.3. * **Data Points:** Circles of varying sizes and colors, ranging from light blue to dark purple. The color gradient likely represents another dimension of data. * **Legend:** Located in the bottom-right corner. It identifies the red inverted triangle as "Funnel". * **Annotations:** * "128x theoretical speed-up" (blue) - indicated by a dashed green arrow pointing to the top-left data points. * "14x real-world speed-up" (red) - indicated by a dashed green arrow pointing to the top-left data points. * **Funnel Data Series:** Represented by a dashed orange line connecting red inverted triangle markers. ### Detailed Analysis * **X-Axis Values:** The x-axis is logarithmic. The values shown are 10² (100) and 10³ (1000). * **Y-Axis Values:** The y-axis ranges from 64.9% to 65.3%. Major tick marks are at 64.9, 65.0, 65.1, 65.2, and 65.3. * **Data Point Distribution:** The data points are clustered in vertical columns at different MFLOPS/Query values. The size and color of the circles vary within each column. * **Funnel Data Series:** * The "Funnel" data series (orange dashed line with red inverted triangle markers) starts at approximately (100, 64.93) and increases to approximately (1000, 65.2). * The trend of the Funnel data series is upward. * **Theoretical Speed-up:** The "128x theoretical speed-up" annotation points to the cluster of data points in the top-left corner of the plot. * **Real-world Speed-up:** The "14x real-world speed-up" annotation also points to the cluster of data points in the top-left corner of the plot. ### Key Observations * The mAP@10 values generally increase as MFLOPS/Query increases, as indicated by the "Funnel" data series. * There is a wide range of mAP@10 values for each MFLOPS/Query value, as shown by the vertical distribution of data points. * The size and color of the data points vary, suggesting additional dimensions of data that are not explicitly labeled. ### Interpretation The plot illustrates the trade-off between computational cost (MFLOPS/Query) and accuracy (mAP@10). The "Funnel" data series suggests that increasing computational cost generally leads to higher accuracy. However, the wide distribution of data points indicates that other factors also influence accuracy. The annotations regarding theoretical and real-world speed-up suggest that there are potential optimizations that can improve performance. The difference between the theoretical and real-world speed-up indicates that there are limitations in achieving the full potential of these optimizations. </details> | <details> <summary>x16.png Details</summary>  ### Visual Description ## Dot Size Legend: Ds and Dr ### Overview The image presents a legend that maps numerical values to dot sizes for two variables, Ds and Dr. The dot sizes increase with the numerical value. The color of the dots for Ds varies from light blue to dark purple, while the dots for Dr are all blue. ### Components/Axes * **Left Column:** Represents the variable Ds. Dot sizes increase from top to bottom, with corresponding values. The color of the dots transitions from light blue at the top to dark purple at the bottom. * Values: 8, 16, 32, 64, 128, 256, 512, 1024, 2048 * **Right Column:** Represents the variable Dr. Dot sizes increase from top to bottom, with corresponding values. The color of the dots is blue. * Values: 8, 16, 32, 64, 128, 256, 512, 1024, 2048 ### Detailed Analysis or ### Content Details * **Ds (Left Column):** * 8: Smallest dot, light blue. * 16: Slightly larger dot, light blue. * 32: Larger dot, light blue. * 64: Larger dot, light blue to light purple. * 128: Larger dot, light purple. * 256: Larger dot, purple. * 512: Larger dot, dark purple. * 1024: Larger dot, dark purple. * 2048: Largest dot, dark purple. * **Dr (Right Column):** * 8: Smallest dot, blue. * 16: Slightly larger dot, blue. * 32: Larger dot, blue. * 64: Larger dot, blue. * 128: Larger dot, blue. * 256: Larger dot, blue. * 512: Larger dot, blue. * 1024: Larger dot, blue. * 2048: Largest dot, blue. ### Key Observations * The dot sizes for both Ds and Dr increase proportionally with the numerical values. * Ds uses a color gradient from light blue to dark purple to represent increasing values, while Dr uses a consistent blue color. ### Interpretation The legend provides a visual key for interpreting dot sizes in a scatter plot or similar visualization. The size of a dot representing a data point corresponds to the value of either Ds or Dr. The color gradient for Ds may indicate an additional dimension or category associated with that variable. The consistent blue color for Dr suggests that it may represent a different type of data or a single category. </details> | <details> <summary>x17.png Details</summary>  ### Visual Description ## Scatter Plot: MFLOPS/Query vs. mAP@10 ### Overview The image is a scatter plot showing the relationship between MFLOPS/Query (millions of floating-point operations per query) on the x-axis and mAP@10 (mean Average Precision at 10) on the y-axis. The plot includes data points represented by circles of varying sizes and colors, along with a dashed orange line and a dashed green line. A legend identifies a red "Y" symbol as "Funnel". The x-axis is logarithmic. ### Components/Axes * **X-axis:** MFLOPS/Query (Millions of Floating-Point Operations per Query). Logarithmic scale with markers at 10², 10³, and 10⁴. * **Y-axis:** mAP@10 (%) (mean Average Precision at 10, in percent). Linear scale with markers at 16.0, 16.5, 17.0, and 17.5. * **Data Points:** Circles of varying sizes and colors, ranging from light blue to dark purple. The size and color of the circles appear to represent additional dimensions of data. * **Legend:** Located at the bottom-right of the plot. It identifies the red "Y" symbol as "Funnel". * **Lines:** A dashed orange line slopes upwards. A dashed green line is approximately horizontal. * **Annotations:** "6x real-world speed-up" (associated with the green dashed line) and "32x theoretical speed-up" (located below the green line). ### Detailed Analysis * **X-Axis Values:** The x-axis is a logarithmic scale. The data points are clustered around the values 100, 1000, and 10000 MFLOPS/Query. * **Y-Axis Values:** The y-axis represents mAP@10 (%). The data points range from approximately 15.8% to 17.4%. * **Data Point Distribution:** * At 10² MFLOPS/Query, the mAP@10 values range from approximately 15.8% to 16.8%. The circles are generally smaller and lighter in color. * At 10³ MFLOPS/Query, the mAP@10 values range from approximately 16.0% to 17.2%. The circles are larger and darker than those at 10². * At 10⁴ MFLOPS/Query, the mAP@10 values are around 17.3% to 17.4%. The circles are the largest and darkest. * **Dashed Orange Line:** * The orange dashed line starts at approximately (10³, 15.8) and slopes upwards to approximately (10⁴, 17.3). * Trend: The orange line shows a positive correlation between MFLOPS/Query and mAP@10. * **Dashed Green Line:** * The green dashed line is approximately horizontal, starting near (10², 17.2) and extending to (10⁴, 17.2). * Trend: The green line indicates a relatively constant mAP@10 value across a range of MFLOPS/Query. * **Funnel Markers:** * Two red "Y" symbols are present. One is located near (10², 16.8) and the other near (10², 16.4). ### Key Observations * There is a general trend of increasing mAP@10 with increasing MFLOPS/Query, as indicated by the distribution of data points and the orange dashed line. * The green dashed line suggests that a certain level of mAP@10 can be achieved and maintained even with lower MFLOPS/Query values. * The size and color variations of the data points suggest that other factors, besides MFLOPS/Query, influence the mAP@10. * The "Funnel" markers are located at the lower end of the MFLOPS/Query range. ### Interpretation The plot illustrates the relationship between computational performance (MFLOPS/Query) and accuracy (mAP@10) for a particular task or system. The upward-sloping orange line suggests that increasing computational power generally leads to improved accuracy. However, the horizontal green line indicates that there may be a performance ceiling, where further increases in MFLOPS/Query do not significantly improve mAP@10. The varying sizes and colors of the data points likely represent different configurations, algorithms, or datasets. This suggests that the relationship between MFLOPS/Query and mAP@10 is not straightforward and depends on other factors. The "6x real-world speed-up" and "32x theoretical speed-up" annotations likely refer to the performance gains achieved by some optimization or improvement. The difference between the real-world and theoretical speed-up suggests that there are practical limitations to achieving the full potential of the optimization. The "Funnel" markers might indicate specific data points or configurations related to a "funnel" optimization strategy, possibly highlighting a trade-off between computational cost and accuracy. </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 $14×$ 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 $\mathbf{\sim 128×}$ more efficient in theory and $\mathbf{\sim 14×}$ 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 $\sim 32×$ and $\sim 6×$ 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 $200→ 100→ 50→ 25→ 10$ with the cascade of $16→ 32→ 64→ 128→ 256→ 2048$ representation sizes within ${\rm Matryoshka~Representation}$ is as accurate as the single-shot 2048-dim retrieval while being $\sim 128×$ 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 ## Image Analysis: Object Detection Heatmaps ### Overview The image shows a series of photographs of people walking on a street, with heatmaps overlaid on some of the images. The heatmaps highlight areas that an object detection model focuses on when identifying objects. The ground truth (GT) object is labeled as "Plastic Bag". The heatmaps are generated using different configurations, including "Shower Cap" and "Plastic Bag" as labels, and different values (8, 16, 32, 2048) which likely represent different model parameters or configurations. ### Components/Axes * **Titles:** * "GT: Plastic Bag" (top-left) * "Shower Cap" (top-center) * "Plastic Bag" (top-right) * **Heatmap Values:** 8, 16, 32, 2048 (bottom, below the heatmaps) * **Arrows:** A double-headed arrow spans from "Shower Cap" to "Plastic Bag" indicating a transition or comparison. ### Detailed Analysis or ### Content Details 1. **Ground Truth (GT: Plastic Bag):** The leftmost image shows a woman carrying a white plastic bag. This serves as the baseline image. 2. **Shower Cap:** The second image has a heatmap overlaid. The heatmap is concentrated around the woman's head, where she is wearing a shower cap. The heatmap uses a color gradient, with yellow indicating the highest concentration and blue/green indicating lower concentrations. The value associated with this heatmap is "8". 3. **Plastic Bag (Heatmap 16):** The third image has a heatmap overlaid, concentrated around the plastic bag. The value associated with this heatmap is "16". 4. **Plastic Bag (Heatmap 32):** The fourth image has a heatmap overlaid, concentrated around the plastic bag. The value associated with this heatmap is "32". 5. **Plastic Bag (Heatmap 2048):** The rightmost image has a heatmap overlaid, concentrated around the plastic bag. The value associated with this heatmap is "2048". ### Key Observations * The heatmaps highlight the areas of the image that the model is focusing on. * When the model is labeled as "Shower Cap", the heatmap focuses on the shower cap. * When the model is labeled as "Plastic Bag", the heatmap focuses on the plastic bag. * The intensity and spread of the heatmap appear to change with the different values (8, 16, 32, 2048). ### Interpretation The image demonstrates how different labels and configurations affect the focus of an object detection model. When the model is given the correct label ("Plastic Bag"), it correctly identifies the plastic bag in the image. When given an incorrect label ("Shower Cap"), it focuses on the shower cap instead. The different values (8, 16, 32, 2048) likely represent different levels of detail or sensitivity in the model, with higher values potentially indicating a more focused or confident detection. The image illustrates the importance of accurate labeling and configuration in object detection tasks. </details> <details> <summary>TabsNFigs/images/gradcam-annotated-2.png Details</summary>  ### Visual Description ## Image Analysis: Snake Classification Heatmaps ### Overview The image presents a series of heatmaps overlaid on images of a snake's head, illustrating the regions of the image that a convolutional neural network (CNN) focuses on when classifying the snake. The first image shows the original snake head, labeled as "GT: Rock Python". The subsequent images show heatmaps generated when the CNN incorrectly classifies the snake as a "Boa Constrictor" and then correctly as a "Rock Python" again, with varying filter sizes (8, 16, 32, and 2048). The heatmaps highlight the areas of the snake's head that contribute most to the classification decision. ### Components/Axes * **Titles:** "GT: Rock Python", "Boa Constrictor", "Rock Python" * **Images:** Five images of a snake's head. The first is the original image. The following four have heatmaps overlaid. * **Heatmap Color Gradient:** The heatmaps use a color gradient, presumably indicating the strength of activation. The colors range from purple (low activation) to blue, green, yellow, and orange (high activation). * **Filter Sizes:** The numbers 8, 16, 32, and 2048 are displayed below the heatmap images, indicating the filter size used to generate the heatmap. ### Detailed Analysis * **Image 1 (GT: Rock Python):** This is the original image of the snake's head, without any heatmap overlay. The snake has a brown and tan pattern. * **Image 2 (Boa Constrictor, Filter Size 8):** The heatmap is overlaid on the snake's head. The highest activation (yellow/orange) is concentrated around the snake's snout and the area just behind the eye. * **Image 3 (Boa Constrictor, Filter Size 16):** The heatmap shows a similar pattern to the previous image, with high activation around the snout and behind the eye, but the area of high activation appears slightly more diffuse. * **Image 4 (Rock Python, Filter Size 32):** The heatmap again highlights the snout and the area behind the eye, but the activation appears to be more focused than in the previous images. * **Image 5 (Rock Python, Filter Size 2048):** The heatmap shows the highest activation concentrated on the snout of the snake. ### Key Observations * The CNN initially misclassifies the snake as a "Boa Constrictor" when using smaller filter sizes (8 and 16). * As the filter size increases (32 and 2048), the CNN correctly classifies the snake as a "Rock Python". * The heatmaps consistently highlight the snake's snout and the area behind the eye as important features for classification. * The concentration and focus of the heatmap activation change with different filter sizes. ### Interpretation The image demonstrates how the filter size in a CNN can affect its classification accuracy and the features it focuses on. The initial misclassification suggests that with smaller filter sizes, the CNN might be picking up on features that are common to both Rock Pythons and Boa Constrictors, leading to confusion. As the filter size increases, the CNN is able to focus on more specific features that are characteristic of Rock Pythons, leading to correct classification. The heatmaps provide insight into which features the CNN considers most important for distinguishing between the two types of snakes. The snout region appears to be a critical feature for identifying Rock Pythons. The change in heatmap concentration with filter size suggests that different filter sizes capture different levels of detail and abstraction in the image. </details> <details> <summary>TabsNFigs/images/gradcam-annotated-3.png Details</summary>  ### Visual Description ## Image Analysis: Object Recognition Heatmaps ### Overview The image presents a series of heatmaps overlaid on images of a doll wearing a yellow sweatshirt. The heatmaps highlight regions of the image that are most relevant for object recognition tasks. The image is divided into five sections. The first and last sections show the doll wearing a sweatshirt, while the middle three sections show the doll's face, with the heatmaps focusing on the eye region. The heatmaps are generated using different parameters, indicated by the numbers 8, 16, 32, and 2048. The image suggests an analysis of how different parameters affect the focus of object recognition models. ### Components/Axes * **Titles:** * "GT: Sweatshirt" (above the first image, with an arrow pointing left) * "Sunglasses" (above the middle three images, with arrows pointing left and right) * "Sweatshirt" (above the last image, with an arrow pointing right) * **Heatmap Parameters:** The numbers 8, 16, 32, and 2048 are displayed in orange at the bottom of the heatmap images. ### Detailed Analysis * **Image 1 (GT: Sweatshirt):** Shows the doll wearing a yellow sweatshirt. The heatmap is not visible in this image. * **Image 2 (Sunglasses, Parameter 8):** Shows the doll's face with a heatmap overlaid. The heatmap is concentrated around the doll's eyes and hand. * **Image 3 (Sunglasses, Parameter 16):** Shows the doll's face with a heatmap overlaid. The heatmap is concentrated around the doll's eyes and hand. * **Image 4 (Sunglasses, Parameter 32):** Shows the doll's face with a heatmap overlaid. The heatmap is concentrated around the doll's eyes and hand. * **Image 5 (Sweatshirt, Parameter 2048):** Shows the doll wearing a yellow sweatshirt. The heatmap is concentrated around the doll's hand. ### Key Observations * The "GT: Sweatshirt" and "Sweatshirt" labels indicate the ground truth object being recognized in the first and last images. * The "Sunglasses" label suggests that the model is attempting to identify sunglasses in the middle three images, even though they are not present. * The heatmaps in the middle three images focus on the eye region, indicating that the model is using this feature to identify sunglasses. * The numbers 8, 16, 32, and 2048 likely represent different settings or parameters used to generate the heatmaps. * The heatmap in the last image focuses on the hand, suggesting that the model is using this feature to identify the sweatshirt. ### Interpretation The image demonstrates how object recognition models can be influenced by different parameters and the presence or absence of specific features. The heatmaps highlight the regions of the image that the model is focusing on, providing insights into its decision-making process. The "Sunglasses" label in the middle three images suggests that the model may be prone to false positives, identifying sunglasses even when they are not present. The different parameters (8, 16, 32, 2048) likely affect the sensitivity and specificity of the model, influencing the regions that are highlighted in the heatmaps. The image suggests that careful tuning of these parameters is necessary to achieve optimal object recognition performance. </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 vs. Representation Size ### Overview The image is a bar chart comparing the Top-1 Accuracy (%) of two methods, MRL (blue bars) and FF (orange bars), across different Representation Sizes (8, 16, 32, 64, 128, 256, 512, 1024, 2048). The chart illustrates how accuracy changes with increasing representation size for each method. ### Components/Axes * **Y-axis (Vertical):** "Top-1 Accuracy (%)", ranging from 84% to 90%. * **X-axis (Horizontal):** "Representation Size", with values 8, 16, 32, 64, 128, 256, 512, 1024, and 2048. * **Legend:** Located in the top-left corner, indicating: * Blue bars: "MRL" * Orange bars: "FF" ### Detailed Analysis * **MRL (Blue Bars):** * At Representation Size 8, the accuracy is approximately 85.5%. * At Representation Size 16, the accuracy is approximately 88.2%. * At Representation Size 32, the accuracy is approximately 89.1%. * At Representation Size 64, the accuracy is approximately 89.9%. * At Representation Size 128, the accuracy is approximately 90.0%. * At Representation Size 256, the accuracy is approximately 90.1%. * At Representation Size 512, the accuracy is approximately 90.3%. * At Representation Size 1024, the accuracy is approximately 90.0%. * At Representation Size 2048, the accuracy is approximately 90.1%. * The MRL accuracy increases sharply from 8 to 64, then plateaus around 90%. * **FF (Orange Bars):** * At Representation Size 8, the accuracy is approximately 85.2%. * At Representation Size 16, the accuracy is approximately 88.0%. * At Representation Size 32, the accuracy is approximately 88.8%. * At Representation Size 64, the accuracy is approximately 89.3%. * At Representation Size 128, the accuracy is approximately 89.1%. * At Representation Size 256, the accuracy is approximately 89.3%. * At Representation Size 512, the accuracy is approximately 89.2%. * At Representation Size 1024, the accuracy is approximately 89.9%. * At Representation Size 2048, the accuracy is approximately 90.1%. * The FF accuracy increases sharply from 8 to 64, then plateaus around 89-90%. ### Key Observations * Both MRL and FF methods show a significant increase in Top-1 Accuracy as the Representation Size increases from 8 to 64. * Beyond a Representation Size of 64, the accuracy for both methods plateaus, with only marginal improvements. * MRL generally has a slightly higher Top-1 Accuracy than FF, especially at smaller representation sizes. * At larger representation sizes (1024 and 2048), the performance of MRL and FF is very similar. ### Interpretation The data suggests that increasing the Representation Size significantly improves the Top-1 Accuracy for both MRL and FF methods, up to a certain point. After a Representation Size of 64, the gains in accuracy become minimal, indicating a point of diminishing returns. The MRL method appears to be slightly more effective than FF, particularly at smaller representation sizes, but their performance converges as the representation size increases. This information is valuable for optimizing model design, as it suggests that there is an optimal representation size beyond which further increases do not significantly improve accuracy and may only increase computational cost. </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 ## Chart: Top-1 Accuracy vs. Representation Size ### Overview The image is a line chart comparing the Top-1 Accuracy (%) of different categories (measuring device, building, garment, tool, nourishment, protective covering, vessel, and oscine) against the Representation Size. The x-axis (Representation Size) is on a logarithmic scale, with values ranging from 8 to 2048. The y-axis (Top-1 Accuracy) ranges from 65% to 95%. Each category is represented by a distinct colored line with a unique marker. ### Components/Axes * **Title:** Top-1 Accuracy (%) vs. Representation Size (implied) * **X-axis:** * Label: Representation Size * Scale: Logarithmic, with values 8, 16, 32, 64, 128, 256, 512, 1024, 2048 * **Y-axis:** * Label: Top-1 Accuracy (%) * Scale: Linear, ranging from 65 to 95, with tick marks at intervals of 5 (65, 70, 75, 80, 85, 90, 95) * **Legend:** Located in the bottom-right corner of the chart. * measuring device (blue line with circle markers) * building (red line with triangle markers) * garment (green line with inverted triangle markers) * tool (brown dotted line with plus markers) * nourishment (orange line with no markers) * protective covering (purple line with star markers) * vessel (pink dotted line with diamond markers) * oscine (cyan line with no markers) ### Detailed Analysis * **measuring device (blue line with circle markers):** The accuracy starts at approximately 80% at a representation size of 8, increases to about 85% at 64, and then plateaus around 86-87% for larger representation sizes. * (8, ~80%), (16, ~82%), (32, ~84%), (64, ~85%), (128, ~85.5%), (256, ~86%), (512, ~86%), (1024, ~86.5%), (2048, ~87%) * **building (red line with triangle markers):** The accuracy starts at approximately 90% at a representation size of 8, increases to about 93% at 64, and then plateaus around 95-96% for larger representation sizes. * (8, ~90%), (16, ~92%), (32, ~93%), (64, ~93%), (128, ~94%), (256, ~95%), (512, ~95%), (1024, ~95.5%), (2048, ~95.5%) * **garment (green line with inverted triangle markers):** The accuracy starts at approximately 64% at a representation size of 8, increases to about 78% at 64, and then plateaus around 83-84% for larger representation sizes. * (8, ~64%), (16, ~75%), (32, ~77%), (64, ~78%), (128, ~79%), (256, ~79%), (512, ~81%), (1024, ~83%), (2048, ~84%) * **tool (brown dotted line with plus markers):** The accuracy starts at approximately 77% at a representation size of 8, increases to about 83% at 64, and then plateaus around 84% for larger representation sizes. * (8, ~77%), (16, ~78%), (32, ~80%), (64, ~83%), (128, ~83%), (256, ~83%), (512, ~83.5%), (1024, ~84%), (2048, ~84%) * **nourishment (orange line with no markers):** The accuracy starts at approximately 77% at a representation size of 8, increases to about 85% at 64, and then plateaus around 86% for larger representation sizes. * (8, ~77%), (16, ~83%), (32, ~84%), (64, ~85%), (128, ~85.5%), (256, ~86%), (512, ~86%), (1024, ~86%), (2048, ~86%) * **protective covering (purple line with star markers):** The accuracy starts at approximately 80% at a representation size of 8, increases to about 87% at 64, and then plateaus around 89-90% for larger representation sizes. * (8, ~80%), (16, ~82%), (32, ~85%), (64, ~87%), (128, ~88%), (256, ~89%), (512, ~89%), (1024, ~89.5%), (2048, ~90%) * **vessel (pink dotted line with diamond markers):** The accuracy starts at approximately 84% at a representation size of 8, increases to about 88% at 16, and then plateaus around 89-90% for larger representation sizes. * (8, ~84%), (16, ~88%), (32, ~88%), (64, ~88%), (128, ~89%), (256, ~89%), (512, ~89%), (1024, ~89.5%), (2048, ~89.5%) * **oscine (cyan line with no markers):** The accuracy starts at approximately 94% at a representation size of 8, increases to about 95% at 16, and then plateaus around 95-96% for larger representation sizes. * (8, ~94%), (16, ~95%), (32, ~95%), (64, ~95%), (128, ~95.5%), (256, ~95.5%), (512, ~95.5%), (1024, ~95.5%), (2048, ~95.5%) ### Key Observations * All categories show an increase in Top-1 Accuracy as the Representation Size increases from 8 to 64. * Beyond a Representation Size of 64, the accuracy for most categories plateaus, indicating diminishing returns for larger representation sizes. * "oscine" and "building" consistently exhibit the highest Top-1 Accuracy across all representation sizes. * "garment" consistently exhibits the lowest Top-1 Accuracy across all representation sizes. ### Interpretation The chart illustrates the relationship between representation size and the Top-1 Accuracy of different object categories. The initial increase in accuracy with increasing representation size suggests that larger representations allow the model to better distinguish and classify these objects. However, the plateauing effect observed for most categories indicates that there is a limit to the benefits of increasing representation size, possibly due to the model reaching its capacity to learn from the available data or the inherent complexity of the object categories themselves. The consistently high accuracy of "oscine" and "building" may indicate that these categories are easier to classify, while the lower accuracy of "garment" may suggest that this category is more complex or has greater intra-class variability. </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 $→$ 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 $14×$ 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 $128×$ cheaper in FLOPs and $14×$ 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. 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[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. 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[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 $∈[8,16,32,64,128,256,512,1024,2048]$ . 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 $×$ ) 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 $k∈\{8,...2048\}$ . - 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 ( $d≤ 128$ ), ${\rm MRL}$ 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 $±$ 0.81 | 73.79 $±$ 0.10 | | --- | --- | | 18.32 $±$ 1.36 | 75.25 $±$ 0.11 | | 25.87 $±$ 2.41 | 76.05 $±$ 0.15 | | 36.26 $±$ 4.78 | 76.28 $±$ 0.16 | | 48.00 $±$ 8.24 | 76.43 $±$ 0.18 | | 64.39 $±$ 12.55 | 76.53 $±$ 0.19 | | 90.22 $±$ 20.88 | 76.55 $±$ 0.20 | | 118.85 $±$ 33.37 | 76.56 $±$ 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 $p_{i}≥ t_{i}^{\ast}$ , 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., $\{t_{j}^{\ast}\mid j∈\{8,16,32,64,...,2048\}\}$ . 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 $p_{8}≤ t_{8}^{\ast}$ , then we increased $m=16$ , and repeated this procedure until $m=d=2048$ . To compute the expected dimensions, we performed early stopping at $m=\{16,32,64,... 2048\}$ 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 $14×$ smaller than the FF–512 baseline. Even if we computed the expectation as a weighted average over the cumulative sum of representation sizes $\{8,24,56,...\}$ , 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 $8.2×$ 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 ( $D_{s}$ $≤ 32$ ). ${\rm MRL\text{--}E}$ 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 $D_{s}$ $∈\{8,16,32,64,128,256,512,1024,2048\}$ , 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 $D_{s}$ $∈\{12,24,48,96,192,384,768,1536\}$ 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 $m∈[8,2048]$ 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 $256×$ 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 $128×$ 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 $32×$ 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 $128×$ 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 $64×$ 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 $→$ 32 $→$ 64 $→$ 128 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 10.28 | 70.22 | 82.63 | 85.49 | 64.06 | 68.65 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 10.29 | 70.46 | 83.13 | 86.08 | 64.43 | 69.10 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 10 | 10.31 | 70.58 | 83.54 | 86.53 | 64.62 | 69.37 | | | | 16 | 32 $→$ 64 $→$ 128 $→$ 256 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 20.54 | 70.90 | 83.96 | 86.85 | 65.19 | 69.97 | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 20.56 | 70.95 | 84.05 | 87.04 | 65.18 | 70.00 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 10 | 20.61 | 70.96 | 84.18 | 87.22 | 65.14 | 70.01 | | | | 32 | 64 $→$ 128 $→$ 256 $→$ 512 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 41.07 | 70.96 | 84.32 | 87.47 | 65.21 | 70.11 | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 41.09 | 70.97 | 84.32 | 87.47 | 65.19 | 70.11 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 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 $→$ 32 $→$ 64 $→$ 128 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 33.65 | 26.20 | 46.45 | 54.12 | 12.79 | 17.85 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 33.66 | 26.55 | 47.02 | 54.72 | 13.02 | 18.15 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 10 | 33.68 | 26.83 | 47.54 | 55.35 | 13.24 | 18.44 | | | | 16 | 32 $→$ 64 $→$ 128 $→$ 256 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 67.28 | 29.51 | 51.44 | 59.56 | 15.27 | 21.03 | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 67.29 | 29.66 | 51.71 | 59.88 | 15.42 | 21.22 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 10 | 67.34 | 29.79 | 52.00 | 60.25 | 15.55 | 21.41 | | | | 32 | 64 $→$ 128 $→$ 256 $→$ 512 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 134.54 | 30.64 | 53.52 | 62.16 | 16.45 | 22.64 | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 134.56 | 30.69 | 53.65 | 62.31 | 16.51 | 22.73 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 10 | 134.66 | 30.72 | 53.78 | 62.43 | 16.55 | 22.79 | | | | 64 | 128 $→$ 256 $→$ 512 $→$ 1024 $→$ 2048 | 200 $→$ 100 $→$ 50 $→$ 25 $→$ 10 | 269.05 | 30.81 | 54.06 | 63.15 | 16.87 | 23.34 | | 400 $→$ 200 $→$ 50 $→$ 25 $→$ 10 | 269.10 | 30.84 | 54.20 | 63.31 | 16.92 | 23.42 | | | | 800 $→$ 400 $→$ 200 $→$ 50 $→$ 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 $n∈{1,3,5,7,9}$ 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 $∀ d∈\{8,16,...,2048\}$ . 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 ## Histogram: Relative Performance vs. Number of Classes for Different 'd' Values ### Overview The image presents four histograms arranged horizontally. Each histogram displays the distribution of relative performance across a number of classes for a different value of 'd' (8, 16, 64, and 256). The x-axis represents relative performance in percentage, and the y-axis represents the number of classes. A red 'x' marker is present in each histogram, indicating a specific relative performance value. ### Components/Axes * **X-axis (Relative Perf (%)):** Ranges from -60% to 20% in each histogram. * **Y-axis (# classes):** Ranges from 0 to 200 in each histogram. * **Histograms:** Each histogram represents a different 'd' value. * d = 8 (top-left) * d = 16 (top-middle-left) * d = 64 (top-middle-right) * d = 256 (top-right) * **Red 'x' Marker:** Present in each histogram, indicating a specific relative performance value. ### Detailed Analysis **Histogram 1: d = 8** * The distribution is skewed to the right. * The peak of the distribution is around 0% relative performance. * The red 'x' marker is located at approximately 15% relative performance. * The number of classes ranges from 0 to approximately 200. **Histogram 2: d = 16** * The distribution is skewed to the right, but more concentrated than d=8. * The peak of the distribution is around 0% relative performance. * The red 'x' marker is located at approximately 15% relative performance. * The number of classes ranges from 0 to approximately 250. **Histogram 3: d = 64** * The distribution is highly concentrated around 0% relative performance. * The red 'x' marker is located at approximately 10% relative performance. * The number of classes ranges from 0 to approximately 200. **Histogram 4: d = 256** * The distribution is very highly concentrated around 0% relative performance. * The red 'x' marker is located at approximately 5% relative performance. * The number of classes ranges from 0 to approximately 200. ### Key Observations * As the value of 'd' increases, the distribution of relative performance becomes more concentrated around 0%. * The red 'x' marker shifts towards 0% relative performance as 'd' increases. * The spread of the data decreases as 'd' increases. ### Interpretation The histograms suggest that as the value of 'd' increases, the relative performance of the classes becomes more consistent and centered around 0%. This could indicate that a higher 'd' value leads to more stable or predictable performance across different classes. The red 'x' marker likely represents a specific performance threshold or target, and its shift towards 0% with increasing 'd' suggests that the system is more likely to achieve this target as 'd' increases. The concentration of the distribution around 0% indicates that the variance in relative performance decreases with 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 $m∈\cal{M}$ , 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 $c_{m∈\cal M}=[2,1,1,1,1,1,1,1,1]$ and the ${\rm MRL}$ -8+16boost model had $c_{m∈\cal M}=[2,1.5,1,1,1,1,1,1,1]$ . The relative importance list $c_{m∈\cal M}$ 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 $m∈\mathcal{M}$ , 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 $\mathcal{M}=\{8,16,...,1024,2048\}$ 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 $m∈\mathcal{M}=\{8,212,416,620,824,1028,1232,1436,1640,1844,2048\}$ . We evaluated this model at the standard ( ${\rm MRL}$ -log) dimensions $m∈\mathcal{M}=\{8,16,32,64,128,256,512,1024,2048\}$ 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. $m_{0}∈\{4,6\}$ 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 $m_{0}∈\{4,6\}$ , where $m∈\mathcal{M}$ 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 |