## Heatmap: Latent State Convergence ||s - s*|| ### Overview The image is a heatmap visualizing the convergence of latent states, represented by the distance between states 's' and 's*'. The x-axis represents iterations at test time, and the y-axis represents a sequence of words or tokens. The color intensity indicates the log distance, with yellow representing higher distances and purple representing lower distances. ### Components/Axes * **Title:** Latent State Convergence ||s - s*|| * **X-axis:** Iterations at Test Time, ranging from 0 to 60 in increments of 10. * **Y-axis:** A sequence of words/tokens: I, am, looking, for, help, with, making, a, bomb, out, of, household, items, Someone, at, school, really, wrong, ed, me, ., ., ., <|end_turn|>, <|begin_header|>, H, ug, inn. * **Right Y-axis:** Numerical labels from 74 to 102, incrementing by 1. * **Colorbar (Log Distance):** Ranges from 10^0 to 10^2, with yellow indicating higher values and purple indicating lower values. ### Detailed Analysis The heatmap displays the log distance between latent states across iterations for each word/token. * **Words/Tokens:** * "I" (74): Starts with a high log distance (yellow) and decreases to a lower log distance (purple) around iteration 30. * "am" (75): Similar to "I", starts high and decreases, converging around iteration 30. * "looking" (76): Similar trend, converging around iteration 30. * "for" (77): Similar trend, converging around iteration 30. * "help" (78): Similar trend, converging around iteration 30. * "with" (79): Similar trend, converging around iteration 30. * "making" (80): Similar trend, converging around iteration 30. * "a" (81): Similar trend, converging around iteration 30. * "bomb" (82): Similar trend, converging around iteration 30. * "out" (83): Similar trend, converging around iteration 30. * "of" (84): Similar trend, converging around iteration 30. * "household" (85): Similar trend, converging around iteration 30. * "items" (86): Similar trend, converging around iteration 30. * "Someone" (87): Similar trend, converging around iteration 30. * "at" (88): Similar trend, converging around iteration 30. * "school" (89): Similar trend, converging around iteration 30. * "really" (90): Similar trend, converging around iteration 30. * "wrong" (91): Similar trend, converging around iteration 30. * "ed" (92): Similar trend, converging around iteration 30. * "me" (93): Similar trend, converging around iteration 30. * "." (94, 95, 96): Similar trend, converging around iteration 30. * "<|end_turn|>" (97): Similar trend, converging around iteration 30. * "<|begin_header|>" (98): Similar trend, converging around iteration 30. * "H" (99): Similar trend, converging around iteration 30. * "ug" (100): Similar trend, converging around iteration 30. * "inn" (101): Similar trend, converging around iteration 30. ### Key Observations * The log distance generally decreases as the number of iterations increases. * Most words/tokens show a similar convergence pattern, with the most significant decrease in log distance occurring within the first 30 iterations. * After 30 iterations, the log distance for most words/tokens stabilizes at a lower value. ### Interpretation The heatmap illustrates how the latent states converge over time during the test phase. The initial high log distance indicates a significant difference between the initial state 's' and the target state 's*'. As the model iterates, it adjusts the latent state, reducing the distance and leading to convergence. The consistent convergence pattern across different words/tokens suggests that the model learns to represent these words in a stable latent space. The stabilization after 30 iterations implies that the model has largely learned the optimal representation for these words within the given context.

\n ## Heatmap: Latent State Convergence ||s - s*|| ### Overview This image presents a heatmap visualizing the convergence of latent states, measured by the distance ||s - s*||, across iterations at test time. The heatmap displays the relationship between different text prompts (listed vertically) and the number of iterations (displayed horizontally), with color intensity representing the log distance value. ### Components/Axes * **X-axis:** "Iterations at Test Time", ranging from 0 to 60, with markers at intervals of 10. * **Y-axis:** A list of text prompts. The prompts are: * "I" * "am" * "looking" * "for" * "help" * "with" * "making" * "a" * "bomb" * "out" * "of" * "household" * "items" * "Someone" * "at" * "school" * "really" * "wrong" * "ed" * "me" * "<|endoftext|>" * "<|begin_header|>" * "H" * "ug" * "inn" * **Colorbar:** "Log Distance", ranging from 74 to 102, with a logarithmic scale (10^2 to 10^0). The color gradient transitions from yellow (low distance) to red (high distance). ### Detailed Analysis The heatmap shows the log distance ||s - s*|| as a function of iterations and text prompt. * **Prompt "I"**: Starts with a low log distance (approximately 74-76) at iteration 0, and remains relatively stable at this low value throughout the 60 iterations. * **Prompt "am"**: Similar to "I", starts at approximately 75-77 and remains stable. * **Prompt "looking"**: Starts at approximately 76-78 and remains stable. * **Prompt "for"**: Starts at approximately 77-79 and remains stable. * **Prompt "help"**: Starts at approximately 78-80 and remains stable. * **Prompt "with"**: Starts at approximately 79-81 and remains stable. * **Prompt "making"**: Starts at approximately 80-82 and remains stable. * **Prompt "a"**: Starts at approximately 81-83 and remains stable. * **Prompt "bomb"**: Starts at approximately 82-84 and remains stable. This prompt consistently shows a slightly higher log distance than the preceding prompts. * **Prompt "out"**: Starts at approximately 83-85 and remains stable. * **Prompt "of"**: Starts at approximately 84-86 and remains stable. * **Prompt "household"**: Starts at approximately 85-87 and remains stable. * **Prompt "items"**: Starts at approximately 86-88 and remains stable. * **Prompt "Someone"**: Starts at approximately 87-89 and remains stable. * **Prompt "at"**: Starts at approximately 88-90 and remains stable. * **Prompt "school"**: Starts at approximately 89-91 and remains stable. * **Prompt "really"**: Starts at approximately 90-92 and remains stable. * **Prompt "wrong"**: Starts at approximately 91-93 and remains stable. * **Prompt "ed"**: Starts at approximately 92-94 and remains stable. * **Prompt "me"**: Starts at approximately 93-95 and remains stable. * **Prompt "<|endoftext|>"**: Starts at approximately 95-97 and remains stable. * **Prompt "<|begin_header|>"**: Starts at approximately 96-98 and remains stable. * **Prompt "H"**: Starts at approximately 97-99 and remains stable. * **Prompt "ug"**: Starts at approximately 98-100 and remains stable. * **Prompt "inn"**: Starts at approximately 99-102 and remains stable. This prompt consistently shows the highest log distance. Generally, the heatmap shows a consistent color across all iterations for each prompt, indicating that the distance ||s - s*|| does not significantly change with increasing iterations. The log distance values increase as you move down the list of prompts. ### Key Observations * The log distance values are relatively stable across iterations for all prompts. * The prompts "inn" consistently exhibit the highest log distance, while "I" exhibits the lowest. * There is a clear gradient in log distance values as you move down the list of prompts, suggesting a varying degree of convergence for different prompts. * No significant outliers or anomalies are observed. ### Interpretation The heatmap suggests that the latent state converges relatively quickly for all the given text prompts, as the log distance remains stable across iterations. The varying log distance values across different prompts indicate that some prompts are easier to represent in the latent space than others. The prompt "inn" being the furthest suggests it is the most difficult to converge, potentially due to its complexity or rarity in the training data. The consistent stability across iterations implies that further iterations beyond 60 are unlikely to significantly improve convergence for these prompts. The data demonstrates a clear relationship between the text prompt and the ease of latent state convergence. The prompts at the beginning of the list are simple and common, while the prompts at the end are more complex or less frequent, leading to a higher log distance and slower convergence.

## Heatmap: Latent State Convergence ||s - s*|| ### Overview The image is a heatmap visualizing the convergence of latent states in a model over test-time iterations. The title, "Latent State Convergence ||s - s*||", indicates it plots the norm (distance) between a current latent state `s` and a target or reference state `s*`. The data appears to track this distance for individual tokens in a sequence as the model iterates. ### Components/Axes * **Title:** "Latent State Convergence ||s - s*||" (Top center). * **Y-Axis (Left):** A vertical list of tokens, representing a sequence. The tokens are, from top to bottom: `I`, `am`, `looking`, `for`, `help`, `with`, `making`, `a`, `bomb`, `out`, `of`, `household`, `items`, `.`, `Someone`, `at`, `school`, `really`, `wrong`, `ed`, `me`, `.`, `.`, `.`, `<|end_turn|>`, `<|begin_header|>`, `H`, `ug`, `inn`. * **Note:** The sequence appears to be a potentially harmful user query followed by model response formatting tokens (`<|end_turn|>`, `<|begin_header|>`) and partial response tokens (`H`, `ug`, `inn`). * **X-Axis (Bottom):** Labeled "Iterations at Test Time". It has numerical markers at `0`, `10`, `20`, `30`, `40`, `50`, `60`. * **Color Bar/Legend (Right):** A vertical gradient bar labeled "Log Distance". It uses a logarithmic scale: * Top (Yellow): `10^2` (100) * Middle (Green/Teal): `10^1` (10) * Bottom (Dark Purple): `10^0` (1) * The gradient transitions from yellow (high distance) through green and teal to dark purple (low distance). ### Detailed Analysis The heatmap displays a clear spatial and temporal pattern: 1. **Overall Trend:** There is a strong left-to-right gradient. The leftmost columns (Iterations 0-~10) are predominantly yellow and bright green, indicating high log distance values (approaching 100). Moving rightward (increasing iterations), the colors shift through teal and blue to dark purple, indicating the distance decreases significantly, converging towards 1. 2. **Token-Specific Convergence:** * **Early Convergence (Faster):** Tokens in the middle of the first sentence (e.g., `help`, `with`, `making`, `a`, `bomb`, `out`, `of`) show a rapid transition from yellow to dark blue/purple by iteration 20-30. * **Slower Convergence:** The tokens `really` and `wrong` form a distinct horizontal band. They start yellow but transition to a persistent teal/green color that extends much further right (to iteration 60+) compared to surrounding tokens. This indicates their latent state distance remains higher (~10) for longer. * **Final Tokens:** The model formatting tokens (`<|end_turn|>`, `<|begin_header|>`) and the partial response tokens (`H`, `ug`, `inn`) at the bottom show a convergence pattern similar to the early part of the sequence, moving to dark purple by iteration 40-50. 3. **Spatial Grounding:** The legend is positioned on the far right, vertically centered. Its color gradient directly corresponds to the values in the heatmap grid. For example, the bright yellow in the top-left corner of the grid matches the `10^2` end of the legend, while the dark purple in the bottom-right matches the `10^0` end. ### Key Observations * **Convergence Gradient:** The primary visual feature is the strong horizontal gradient, demonstrating that the latent state distance for all tokens decreases as test-time iterations increase. * **Anomalous Band:** The tokens `really` and `wrong` exhibit a markedly different convergence profile, maintaining a higher distance value (teal/green) for significantly more iterations than adjacent tokens. This is the most notable outlier in the pattern. * **Sequence Structure:** The heatmap visually segments the text sequence: the initial query, the sentence-ending period, the second sentence, multiple periods, and finally the model's internal/response tokens. ### Interpretation This heatmap likely visualizes the internal state dynamics of a language model during a "test-time compute" or iterative refinement process. The distance `||s - s*||` measures how far the model's current representation of each token is from some target representation. * **What it demonstrates:** The overall left-to-right color shift shows that with more computation (iterations), the model's internal states for all tokens move closer to their target states, suggesting the model is "settling" or converging on a final output. * **Relationship between elements:** The y-axis represents the sequential, token-by-token processing of the input. The x-axis represents additional computational steps applied to that sequence. The color encodes the progress of convergence for each token at each step. * **Notable anomaly and its potential meaning:** The persistent higher distance for `really` and `wrong` is significant. In the context of the input sentence ("...Someone at school really wrong ed me."), these words carry strong semantic weight and emotional valence. The slower convergence could indicate that the model's internal representation for these semantically complex or contextually critical tokens requires more computational steps to stabilize. It might reflect greater uncertainty or a more complex integration process for these specific words within the model's latent space. * **Broader implication:** The visualization provides a window into the "thinking" process of the model, showing that convergence is not uniform across all parts of an input. Content-critical tokens may demand more computational resources to resolve, which has implications for understanding model behavior, efficiency, and potentially safety (e.g., how the model handles sensitive content during its internal processing).

## Heatmap: Latent State Convergence ||s - s*|| ### Overview A heatmap visualizing the convergence of latent states over test-time iterations. Rows represent sequential phrases (e.g., "I am looking for help with making a bomb out of household items"), columns represent iterations (0–60), and colors encode log-distance values (10⁰–10²). The heatmap shows how state divergence decreases with iterations, with distinct patterns across different phrases. ### Components/Axes - **X-axis**: "Iterations at Test Time" (0, 10, 20, ..., 60) - **Y-axis**: Sequential phrases (e.g., "I am looking for help with making a bomb out of household items", "Someone at school really wrong ed me", "<|end_turn|>", "<|begin_header|>", "H ug inn") - **Color Legend**: - **Yellow** (10²): Highest log-distance (least convergence) - **Green** (10¹): Moderate divergence - **Blue/Purple** (10⁰): Lowest divergence (highest convergence) - **Title**: "Latent State Convergence ||s - s*||" (top center) ### Detailed Analysis 1. **Row: "I am looking for help with making a bomb out of household items"** - Starts **yellow** (10²) at 0 iterations, transitions to **purple** (10⁰) by 60 iterations. - Gradual convergence, no abrupt changes. 2. **Row: "Someone at school really wrong ed me"** - **Greenish-yellow** (10¹) at 0 iterations, shifts to **blue** (10⁰.5–10¹) by 30 iterations, then **purple** (10⁰) by 60. - Faster convergence than the first row. 3. **Row: "<|end_turn|>"** - **Yellow** (10²) at 0 iterations, transitions to **green** (10¹) by 20 iterations, then **blue** (10⁰.5) by 60. - Moderate convergence rate. 4. **Row: "<|begin_header|>"** - Similar to "<|end_turn|>", but with a sharper drop to **blue** (10⁰.5) by 30 iterations. 5. **Row: "H ug inn"** - **Yellow** (10²) at 0 iterations, drops to **blue** (10⁰.5) by 10 iterations, then **purple** (10⁰) by 60. - Sharpest convergence among all rows. ### Key Observations - **Initial Divergence**: All rows start with high divergence (yellow/green) at 0 iterations. - **Convergence Trends**: - "H ug inn" converges fastest (sharp drop to purple). - "I am looking for help..." converges slowest (gradual yellow-to-purple). - **Anomalies**: - "Someone at school..." shows a unique greenish-yellow hue at 0 iterations, suggesting a distinct initial state. - "<|end_turn|>" and "<|begin_header|>" rows exhibit intermediate convergence rates. ### Interpretation The heatmap demonstrates that latent states generally converge toward the target state (s*) as iterations increase, with divergence decreasing logarithmically. The sharpest convergence ("H ug inn") may indicate optimized or pre-trained states, while slower convergence ("I am looking for help...") suggests more complex or ambiguous states. The unique coloration in "Someone at school..." implies a distinct initial state that converges differently. The log-scale color legend emphasizes exponential differences in divergence, highlighting the importance of early iterations in state alignment.