## Heatmap: Coverage of Different Generation and Refinement Methods ### Overview The image is a heatmap displaying the coverage between different combinations of generation and refinement methods. The methods include "Direct Generation," "Repeated Sampling," and "Refinement," each with variations "P," "C," and "PC." The heatmap uses a color gradient to represent coverage values, ranging from light (low coverage) to dark (high coverage). The matrix is symmetrical along the diagonal, where each method compared to itself has a coverage of 1.00. ### Components/Axes * **Rows and Columns:** The rows and columns represent the different generation and refinement methods: * Direct Generation P * Direct Generation C * Direct Generation PC * Repeated Sampling P * Repeated Sampling C * Repeated Sampling PC * Refinement P * Refinement C * Refinement PC * **Color Scale (Legend):** Located on the right side of the heatmap. * Dark Red: Represents a coverage of 1.0 * Light Yellow: Represents a coverage of 0.0 * Intermediate colors represent coverage values between 0.0 and 1.0. * **Title:** "QwQ-32B" at the bottom center of the heatmap. ### Detailed Analysis The heatmap displays the coverage between different combinations of generation and refinement methods. The values range from 0.32 to 1.00. * **Direct Generation P:** * Coverage with Direct Generation P: 1.00 * Coverage with Direct Generation C: 0.53 * Coverage with Direct Generation PC: 0.37 * Coverage with Repeated Sampling P: 0.68 * Coverage with Repeated Sampling C: 0.71 * Coverage with Repeated Sampling PC: 0.74 * Coverage with Refinement P: 0.68 * Coverage with Refinement C: 0.71 * Coverage with Refinement PC: 0.74 * **Direct Generation C:** * Coverage with Direct Generation P: 0.69 * Coverage with Direct Generation C: 1.00 * Coverage with Direct Generation PC: 0.45 * Coverage with Repeated Sampling P: 0.79 * Coverage with Repeated Sampling C: 0.93 * Coverage with Repeated Sampling PC: 0.86 * Coverage with Refinement P: 0.72 * Coverage with Refinement C: 0.86 * Coverage with Refinement PC: 0.86 * **Direct Generation PC:** * Coverage with Direct Generation P: 0.61 * Coverage with Direct Generation C: 0.57 * Coverage with Direct Generation PC: 1.00 * Coverage with Repeated Sampling P: 0.70 * Coverage with Repeated Sampling C: 0.78 * Coverage with Repeated Sampling PC: 0.91 * Coverage with Refinement P: 0.61 * Coverage with Refinement C: 0.83 * Coverage with Refinement PC: 0.78 * **Repeated Sampling P:** * Coverage with Direct Generation P: 0.58 * Coverage with Direct Generation C: 0.51 * Coverage with Direct Generation PC: 0.36 * Coverage with Repeated Sampling P: 1.00 * Coverage with Repeated Sampling C: 0.73 * Coverage with Repeated Sampling PC: 0.76 * Coverage with Refinement P: 0.64 * Coverage with Refinement C: 0.69 * Coverage with Refinement PC: 0.69 * **Repeated Sampling C:** * Coverage with Direct Generation P: 0.50 * Coverage with Direct Generation C: 0.50 * Coverage with Direct Generation PC: 0.33 * Coverage with Repeated Sampling P: 0.61 * Coverage with Repeated Sampling C: 1.00 * Coverage with Repeated Sampling PC: 0.80 * Coverage with Refinement P: 0.65 * Coverage with Refinement C: 0.72 * Coverage with Refinement PC: 0.70 * **Repeated Sampling PC:** * Coverage with Direct Generation P: 0.49 * Coverage with Direct Generation C: 0.44 * Coverage with Direct Generation PC: 0.37 * Coverage with Repeated Sampling P: 0.60 * Coverage with Repeated Sampling C: 0.75 * Coverage with Repeated Sampling PC: 1.00 * Coverage with Refinement P: 0.54 * Coverage with Refinement C: 0.70 * Coverage with Refinement PC: 0.63 * **Refinement P:** * Coverage with Direct Generation P: 0.59 * Coverage with Direct Generation C: 0.48 * Coverage with Direct Generation PC: 0.32 * Coverage with Repeated Sampling P: 0.66 * Coverage with Repeated Sampling C: 0.80 * Coverage with Repeated Sampling PC: 0.70 * Coverage with Refinement P: 1.00 * Coverage with Refinement C: 0.73 * Coverage with Refinement PC: 0.73 * **Refinement C:** * Coverage with Direct Generation P: 0.47 * Coverage with Direct Generation C: 0.44 * Coverage with Direct Generation PC: 0.33 * Coverage with Repeated Sampling P: 0.54 * Coverage with Repeated Sampling C: 0.68 * Coverage with Repeated Sampling PC: 0.70 * Coverage with Refinement P: 0.56 * Coverage with Refinement C: 1.00 * Coverage with Refinement PC: 0.72 * **Refinement PC:** * Coverage with Direct Generation P: 0.50 * Coverage with Direct Generation C: 0.45 * Coverage with Direct Generation PC: 0.32 * Coverage with Repeated Sampling P: 0.55 * Coverage with Repeated Sampling C: 0.68 * Coverage with Repeated Sampling PC: 0.64 * Coverage with Refinement P: 0.57 * Coverage with Refinement C: 0.73 * Coverage with Refinement PC: 1.00 ### Key Observations * The diagonal elements are all 1.00, indicating perfect coverage when a method is compared to itself. * Direct Generation C has the highest coverage with Repeated Sampling C (0.93) and Refinement C (0.86). * Direct Generation PC has the highest coverage with Repeated Sampling PC (0.91) and Refinement C (0.83). * Refinement C and Refinement PC show relatively high coverage with each other (0.73 and 0.72, respectively). * The lowest coverage values are generally observed between Direct Generation PC and other methods. ### Interpretation The heatmap provides a comparative analysis of the coverage achieved by different generation and refinement methods. The high coverage values along the diagonal are expected, as they represent the self-comparison of each method. The off-diagonal values indicate the degree of overlap or similarity between the outputs of different methods. The higher coverage between Direct Generation C and Repeated Sampling/Refinement C suggests that these methods produce more similar results. Conversely, the lower coverage between Direct Generation PC and other methods may indicate that this method generates more diverse or distinct outputs. The data suggests that the choice of generation and refinement methods can significantly impact the coverage and similarity of the generated content. Researchers and practitioners can use this information to select the most appropriate combination of methods for their specific needs and objectives.

\n ## Heatmap: Correlation Matrix of Sampling Methods ### Overview This image presents a heatmap displaying a correlation matrix between different sampling methods used in a process. The methods are variations of Direct Generation, Repeated Sampling, and Refinement, each with options for 'P' (presumably a parameter or process) and 'C' (another parameter or process), and 'PC' (potentially a combination). The color intensity represents the correlation coefficient, with a scale ranging from 0.0 to 1.0. ### Components/Axes * **X-axis:** Lists the sampling methods: "Direct Generation\_P", "Direct Generation\_C", "Direct Generation\_PC", "Repeated Sampling\_P", "Repeated Sampling\_C", "Repeated Sampling\_PC", "Refinement\_P", "Refinement\_C", "Refinement\_PC". * **Y-axis:** Lists the same sampling methods as the X-axis, creating a square matrix. * **Color Scale (Right Side):** A vertical color bar indicates the correlation coefficient. * 0.0 is represented by a light color (almost white). * 1.0 is represented by a dark red color. * 0.5 is represented by a mid-tone color. * **Cell Values:** Each cell in the matrix displays a numerical value representing the correlation coefficient between the corresponding X and Y axis sampling methods. * **Label (Bottom):** "Qwq-32B" - likely an identifier for the dataset or experiment. ### Detailed Analysis The heatmap displays correlation coefficients between nine sampling methods. The values range from approximately 0.32 to 1.00. Here's a breakdown of the values, row by row, referencing the color intensity: * **Direct Generation\_P:** * Direct Generation\_P: 1.00 * Direct Generation\_C: 0.53 * Direct Generation\_PC: 0.37 * Repeated Sampling\_P: 0.68 * Repeated Sampling\_C: 0.71 * Repeated Sampling\_PC: 0.74 * Refinement\_P: 0.68 * Refinement\_C: 0.71 * Refinement\_PC: 0.74 * **Direct Generation\_C:** * Direct Generation\_P: 0.69 * Direct Generation\_C: 1.00 * Direct Generation\_PC: 0.45 * Repeated Sampling\_P: 0.79 * Repeated Sampling\_C: 0.93 * Repeated Sampling\_PC: 0.86 * Refinement\_P: 0.72 * Refinement\_C: 0.86 * Refinement\_PC: 0.86 * **Direct Generation\_PC:** * Direct Generation\_P: 0.61 * Direct Generation\_C: 0.57 * Direct Generation\_PC: 1.00 * Repeated Sampling\_P: 0.70 * Repeated Sampling\_C: 0.78 * Repeated Sampling\_PC: 0.91 * Refinement\_P: 0.61 * Refinement\_C: 0.83 * Refinement\_PC: 0.78 * **Repeated Sampling\_P:** * Direct Generation\_P: 0.58 * Direct Generation\_C: 0.51 * Direct Generation\_PC: 0.36 * Repeated Sampling\_P: 1.00 * Repeated Sampling\_C: 0.73 * Repeated Sampling\_PC: 0.76 * Refinement\_P: 0.64 * Refinement\_C: 0.69 * Refinement\_PC: 0.69 * **Repeated Sampling\_C:** * Direct Generation\_P: 0.50 * Direct Generation\_C: 0.50 * Direct Generation\_PC: 0.33 * Repeated Sampling\_P: 0.61 * Repeated Sampling\_C: 1.00 * Repeated Sampling\_PC: 0.80 * Refinement\_P: 0.65 * Refinement\_C: 0.72 * Refinement\_PC: 0.70 * **Repeated Sampling\_PC:** * Direct Generation\_P: 0.49 * Direct Generation\_C: 0.44 * Direct Generation\_PC: 0.37 * Repeated Sampling\_P: 0.60 * Repeated Sampling\_C: 0.75 * Repeated Sampling\_PC: 1.00 * Refinement\_P: 0.54 * Refinement\_C: 0.70 * Refinement\_PC: 0.63 * **Refinement\_P:** * Direct Generation\_P: 0.59 * Direct Generation\_C: 0.48 * Direct Generation\_PC: 0.32 * Repeated Sampling\_P: 0.66 * Repeated Sampling\_C: 0.80 * Repeated Sampling\_PC: 0.70 * Refinement\_P: 1.00 * Refinement\_C: 0.73 * Refinement\_PC: 0.73 * **Refinement\_C:** * Direct Generation\_P: 0.47 * Direct Generation\_C: 0.44 * Direct Generation\_PC: 0.33 * Repeated Sampling\_P: 0.54 * Repeated Sampling\_C: 0.68 * Repeated Sampling\_PC: 0.70 * Refinement\_P: 0.56 * Refinement\_C: 1.00 * Refinement\_PC: 0.72 * **Refinement\_PC:** * Direct Generation\_P: 0.50 * Direct Generation\_C: 0.45 * Direct Generation\_PC: 0.32 * Repeated Sampling\_P: 0.55 * Repeated Sampling\_C: 0.68 * Repeated Sampling\_PC: 0.64 * Refinement\_P: 0.57 * Refinement\_C: 0.73 * Refinement\_PC: 1.00 ### Key Observations * **Perfect Self-Correlation:** All methods have a correlation of 1.00 with themselves (diagonal elements). * **Strong Correlation within Method Type:** Within each sampling method type (Direct Generation, Repeated Sampling, Refinement), the 'P', 'C', and 'PC' variations show relatively high correlations, generally above 0.6. * **Highest Correlations:** The highest correlations (excluding self-correlation) are observed between "Direct Generation\_C" and "Repeated Sampling\_C" (0.93), and between "Direct Generation\_PC" and "Repeated Sampling\_PC" (0.91). * **Lowest Correlations:** The lowest correlations are generally found between the 'PC' variations of different methods (e.g., Direct Generation\_PC and Repeated Sampling\_P, 0.36). * **Symmetry:** The matrix is symmetrical, indicating that the correlation between method A and method B is the same as between method B and method A. ### Interpretation This correlation matrix suggests that the different sampling methods are related, but not perfectly interchangeable. The strong correlations within each method type (Direct, Repeated, Refinement) indicate that the 'P', 'C', and 'PC' variations are capturing similar information. The higher correlations between 'C' variations suggest that the 'C' parameter or process is a dominant factor in determining the similarity between these methods. The lower correlations between 'PC' variations of different methods suggest that the combination of 'P' and 'C' introduces more diversity in the sampling process, leading to less overlap in the information captured. The dataset identifier "Qwq-32B" implies this data is from a specific model or experiment, and the correlations observed are specific to that context. The heatmap provides valuable insights into the relationships between these sampling methods, which can be used to inform decisions about which methods to use in combination or to select the most appropriate method for a given task. Further investigation could explore the meaning of the 'P' and 'C' parameters to understand why certain correlations are stronger than others.

## Heatmap: Coverage Similarity Between Generation Methods (QwQ-32B) ### Overview This image is a 9x9 heatmap visualizing the "Coverage" similarity or overlap between nine different text generation methods or strategies. The methods are grouped into three categories: Direct Generation, Repeated Sampling, and Refinement, each with three variants (P, C, PC). The heatmap uses a color gradient from light beige (0.0) to dark red (1.0) to represent the coverage value, which is also printed numerically in each cell. The diagonal cells, representing a method compared to itself, all have a value of 1.00. ### Components/Axes * **Chart Type:** Heatmap (symmetrical matrix). * **X-Axis (Top):** Labels are rotated 45 degrees. From left to right: 1. Direct Generation P 2. Direct Generation C 3. Direct Generation PC 4. Repeated Sampling P 5. Repeated Sampling C 6. Repeated Sampling PC 7. Refinement P 8. Refinement C 9. Refinement PC * **Y-Axis (Left):** Labels are rotated 45 degrees. From top to bottom, the same nine categories as the X-axis. * **Legend/Color Scale:** Located on the right side of the chart. It is a vertical bar labeled "Coverage" with a gradient from light beige at the bottom (value `0.0`) to dark red at the top (value `1.0`). A midpoint marker indicates `0.5`. * **Footer Label:** The text "QwQ-32B" is centered at the very bottom of the image, likely indicating the model or dataset used. ### Detailed Analysis The matrix is symmetric (e.g., value at [Row: Direct Generation C, Column: Repeated Sampling P] equals value at [Row: Repeated Sampling P, Column: Direct Generation C]). Below is the reconstructed data table. Values are read directly from the cells. | Method (Row \ Column) | Direct Gen P | Direct Gen C | Direct Gen PC | Repeated Samp P | Repeated Samp C | Repeated Samp PC | Refinement P | Refinement C | Refinement PC | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Direct Generation P** | **1.00** | 0.53 | 0.37 | 0.68 | 0.71 | 0.74 | 0.68 | 0.71 | 0.74 | | **Direct Generation C** | 0.69 | **1.00** | 0.45 | 0.79 | 0.93 | 0.86 | 0.72 | 0.86 | 0.86 | | **Direct Generation PC** | 0.61 | 0.57 | **1.00** | 0.70 | 0.78 | 0.91 | 0.61 | 0.83 | 0.78 | | **Repeated Sampling P** | 0.58 | 0.51 | 0.36 | **1.00** | 0.73 | 0.76 | 0.64 | 0.69 | 0.69 | | **Repeated Sampling C** | 0.50 | 0.50 | 0.33 | 0.61 | **1.00** | 0.80 | 0.65 | 0.72 | 0.70 | | **Repeated Sampling PC** | 0.49 | 0.44 | 0.37 | 0.60 | 0.75 | **1.00** | 0.54 | 0.70 | 0.63 | | **Refinement P** | 0.59 | 0.48 | 0.32 | 0.66 | 0.80 | 0.70 | **1.00** | 0.73 | 0.73 | | **Refinement C** | 0.47 | 0.44 | 0.33 | 0.54 | 0.68 | 0.70 | 0.56 | **1.00** | 0.72 | | **Refinement PC** | 0.50 | 0.45 | 0.32 | 0.55 | 0.68 | 0.64 | 0.57 | 0.73 | **1.00** | **Trend Verification & Color Correlation:** * **Diagonal Trend:** All diagonal cells are the darkest red, corresponding to the maximum value of 1.00, confirming perfect self-similarity. * **High-Value Clusters:** The darkest red off-diagonal cells indicate high coverage. Notable examples: * `Direct Generation C` vs. `Repeated Sampling C` (0.93) - Very high similarity. * `Direct Generation PC` vs. `Repeated Sampling PC` (0.91) - Very high similarity. * `Direct Generation C` vs. `Repeated Sampling PC` (0.86) and `Refinement C` (0.86). * **Low-Value Clusters:** The lightest beige/orange cells indicate low coverage. Notable examples: * `Direct Generation PC` vs. `Refinement P` (0.32) and `Refinement PC` (0.32). * `Direct Generation P` vs. `Direct Generation PC` (0.37). * `Repeated Sampling PC` vs. `Direct Generation PC` (0.37). * **General Pattern:** Methods within the same category (e.g., Direct Generation C vs. Direct Generation PC) often have moderate to high similarity. Cross-category comparisons (e.g., Direct Generation vs. Refinement) show more variability. ### Key Observations 1. **Highest Inter-Method Similarity:** The pair `Direct Generation C` and `Repeated Sampling C` has the highest off-diagonal coverage (0.93), suggesting these two methods produce highly overlapping outputs. 2. **Lowest Inter-Method Similarity:** The pair `Direct Generation PC` and `Refinement P` (also `Refinement PC`) has the lowest coverage (0.32), indicating these methods have the least overlap in their outputs. 3. **"PC" Variant Behavior:** The "PC" variants (Direct Generation PC, Repeated Sampling PC, Refinement PC) tend to have lower coverage scores when compared to "P" or "C" variants from other categories, suggesting they may be more distinct strategies. 4. **Refinement Category Consistency:** The Refinement methods (P, C, PC) show relatively consistent, moderate coverage scores when compared to each other (0.72-0.73), but lower scores when compared to Direct Generation methods. ### Interpretation This heatmap quantifies the similarity in "coverage" (likely meaning the set of problems solved, outputs generated, or knowledge demonstrated) between different prompting or generation strategies for the QwQ-32B model. * **What it demonstrates:** The data suggests that **methodology significantly impacts output characteristics**. Strategies sharing a core approach (e.g., all "C" variants) have high mutual coverage, meaning they tend to succeed on similar tasks. The very high similarity between `Direct Generation C` and `Repeated Sampling C` implies that for the "C" condition, simply repeating the sampling process doesn't fundamentally change *what* the model can produce compared to a single direct generation. * **Relationships between elements:** The matrix reveals a structure where the "C" and "PC" variants form stronger clusters with their counterparts in other categories than the "P" variants do. This could indicate that the "C" (perhaps "Constrained" or "Chain-of-thought") and "PC" (perhaps "Plan-and-Code") conditions impose a stronger, more consistent behavioral signature on the model than the "P" (perhaps "Plain") condition. * **Notable Anomaly/Insight:** The consistently low coverage between `Direct Generation PC` and the `Refinement` methods is striking. It suggests that the "Plan-and-Code" direct generation approach explores a solution space that is largely disjoint from the space explored by refinement-based methods. This could be because refinement methods start from an initial draft and improve it, while direct PC generation attempts to produce a complete, structured solution in one pass, leading to fundamentally different types of outputs or error profiles. In essence, this chart is a map of strategic similarity. It shows that not all generation methods are created equal; they occupy different "niches" in terms of the problems they can solve, with some being nearly interchangeable (high coverage) and others being complementary (low coverage). This information is crucial for designing ensemble methods or choosing the right strategy for a specific task.

# Technical Document Extraction: Heatmap Analysis ## 1. Axis Labels and Titles - **X-axis**: `QwQ-32B` (repeated across all column headers) - **Y-axis**: - `Direct Generation P` - `Direct Generation C` - `Direct Generation PC` - `Repeated Sampling P` - `Repeated Sampling C` - `Repeated Sampling PC` - `Refinement P` - `Refinement C` - `Refinement PC` ## 2. Legend - **Color Scale**: - `0.0` (light orange) to `1.0` (dark red) - Gradient represents coverage values ## 3. Data Table Structure A 10x10 matrix where: - **Rows**: Methods (e.g., `Direct Generation P`, `Repeated Sampling C`) - **Columns**: Methods appended with `QwQ-32B` (e.g., `Direct Generation P QwQ-32B`) - **Values**: Coverage scores (0.0–1.0) ## 4. Key Trends - **Diagonal Dominance**: All diagonal cells (same method vs. itself) = `1.00` (dark red). - **High Coverage Clusters**: - `Direct Generation C` and `Refinement C` show high coverage (`0.86–0.93`) with `Repeated Sampling C QwQ-32B`. - `Repeated Sampling PC` and `Refinement PC` exhibit strong mutual coverage (`0.70–0.75`). - **Low Coverage**: - `Direct Generation P` vs. `Repeated Sampling P QwQ-32B` = `0.58`. - `Refinement P` vs. `Repeated Sampling P QwQ-32B` = `0.66`. ## 5. Spatial Grounding - **Legend Position**: Right side of the heatmap. - **Color Consistency**: - `0.32` (light orange) matches `Repeated Sampling P` vs. `Direct Generation PC QwQ-32B`. - `0.91` (dark red) matches `Repeated Sampling C` vs. `Direct Generation PC QwQ-32B`. ## 6. Full Data Table | Method (Y-axis) | Direct Generation P QwQ-32B | Direct Generation C QwQ-32B | Direct Generation PC QwQ-32B | Repeated Sampling P QwQ-32B | Repeated Sampling C QwQ-32B | Repeated Sampling PC QwQ-32B | Refinement P QwQ-32B | Refinement C QwQ-32B | Refinement PC QwQ-32B | |--------------------------|-----------------------------|-----------------------------|------------------------------|-----------------------------|-----------------------------|------------------------------|----------------------|----------------------|-----------------------| | **Direct Generation P** | 1.00 | 0.53 | 0.37 | 0.68 | 0.71 | 0.74 | 0.68 | 0.71 | 0.74 | | **Direct Generation C** | 0.69 | 1.00 | 0.45 | 0.79 | 0.93 | 0.86 | 0.72 | 0.86 | 0.86 | | **Direct Generation PC** | 0.61 | 0.57 | 1.00 | 0.70 | 0.78 | 0.91 | 0.61 | 0.83 | 0.78 | | **Repeated Sampling P** | 0.58 | 0.51 | 0.36 | 1.00 | 0.73 | 0.76 | 0.64 | 0.69 | 0.69 | | **Repeated Sampling C** | 0.50 | 0.50 | 0.33 | 0.61 | 1.00 | 0.80 | 0.65 | 0.72 | 0.70 | | **Repeated Sampling PC** | 0.49 | 0.44 | 0.37 | 0.60 | 0.75 | 1.00 | 0.54 | 0.70 | 0.63 | | **Refinement P** | 0.59 | 0.48 | 0.32 | 0.66 | 0.80 | 0.70 | 1.00 | 0.73 | 0.73 | | **Refinement C** | 0.47 | 0.44 | 0.33 | 0.54 | 0.68 | 0.70 | 0.56 | 1.00 | 0.72 | | **Refinement PC** | 0.50 | 0.45 | 0.32 | 0.55 | 0.68 | 0.64 | 0.57 | 0.73 | 1.00 | ## 7. Observations - **Method Similarity**: Methods with similar prefixes (e.g., `Direct Generation` vs. `Refinement`) show moderate coverage (`0.44–0.74`). - **QwQ-32B Impact**: Coverage generally decreases when compared to `QwQ-32B` variants, except for diagonal entries. ## 8. Language Notes - All text is in English. No non-English content detected.