\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.