## Heatmap: Coverage of Different Generation Methods ### Overview The image is a heatmap displaying the coverage between different text generation methods using the "gpt-03-mini" model. The heatmap shows the pairwise coverage scores, with darker red shades indicating higher coverage and lighter shades indicating lower coverage. The methods compared are Direct Generation, Repeated Sampling, and Refinement, each using P, C, and PC configurations. ### Components/Axes * **X-axis (Columns):** Text generation 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 (Rows):** Text generation 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. The scale ranges from 0.0 (lightest shade) to 1.0 (darkest shade), representing the "Coverage" score. * **Model Name:** "gpt-03-mini" is printed at the bottom center of the heatmap. ### Detailed Analysis The heatmap presents a matrix of coverage scores between different text generation methods. Each cell contains a numerical value representing the coverage between the method on the Y-axis (row) and the method on the X-axis (column). The diagonal elements, where the row and column methods are the same, have a coverage of 1.00, indicated by the darkest red color. Here's a breakdown of the coverage scores: * **Direct Generation P:** * Coverage with Direct Generation P: 1.00 * Coverage with Direct Generation C: 0.74 * Coverage with Direct Generation PC: 0.75 * Coverage with Repeated Sampling P: 0.80 * Coverage with Repeated Sampling C: 0.89 * Coverage with Repeated Sampling PC: 0.84 * Coverage with Refinement P: 0.74 * Coverage with Refinement C: 0.81 * Coverage with Refinement PC: 0.75 * **Direct Generation C:** * Coverage with Direct Generation P: 0.61 * Coverage with Direct Generation C: 1.00 * Coverage with Direct Generation PC: 0.71 * Coverage with Repeated Sampling P: 0.68 * Coverage with Repeated Sampling C: 0.89 * Coverage with Repeated Sampling PC: 0.83 * Coverage with Refinement P: 0.76 * Coverage with Refinement C: 0.80 * Coverage with Refinement PC: 0.70 * **Direct Generation PC:** * Coverage with Direct Generation P: 0.69 * Coverage with Direct Generation C: 0.79 * Coverage with Direct Generation PC: 1.00 * Coverage with Repeated Sampling P: 0.73 * Coverage with Repeated Sampling C: 0.91 * Coverage with Repeated Sampling PC: 0.84 * Coverage with Refinement P: 0.73 * Coverage with Refinement C: 0.81 * Coverage with Refinement PC: 0.80 * **Repeated Sampling P:** * Coverage with Direct Generation P: 0.68 * Coverage with Direct Generation C: 0.71 * Coverage with Direct Generation PC: 0.68 * Coverage with Repeated Sampling P: 1.00 * Coverage with Repeated Sampling C: 0.87 * Coverage with Repeated Sampling PC: 0.88 * Coverage with Refinement P: 0.80 * Coverage with Refinement C: 0.75 * Coverage with Refinement PC: 0.69 * **Repeated Sampling C:** * Coverage with Direct Generation P: 0.55 * Coverage with Direct Generation C: 0.67 * Coverage with Direct Generation PC: 0.62 * Coverage with Repeated Sampling P: 0.64 * Coverage with Repeated Sampling C: 1.00 * Coverage with Repeated Sampling PC: 0.80 * Coverage with Refinement P: 0.65 * Coverage with Refinement C: 0.78 * Coverage with Refinement PC: 0.69 * **Repeated Sampling PC:** * Coverage with Direct Generation P: 0.55 * Coverage with Direct Generation C: 0.66 * Coverage with Direct Generation PC: 0.61 * Coverage with Repeated Sampling P: 0.68 * Coverage with Repeated Sampling C: 0.85 * Coverage with Repeated Sampling PC: 1.00 * Coverage with Refinement P: 0.67 * Coverage with Refinement C: 0.76 * Coverage with Refinement PC: 0.67 * **Refinement P:** * Coverage with Direct Generation P: 0.61 * Coverage with Direct Generation C: 0.75 * Coverage with Direct Generation PC: 0.66 * Coverage with Repeated Sampling P: 0.77 * Coverage with Repeated Sampling C: 0.85 * Coverage with Repeated Sampling PC: 0.84 * Coverage with Refinement P: 1.00 * Coverage with Refinement C: 0.75 * Coverage with Refinement PC: 0.71 * **Refinement C:** * Coverage with Direct Generation P: 0.56 * Coverage with Direct Generation C: 0.67 * Coverage with Direct Generation PC: 0.62 * Coverage with Repeated Sampling P: 0.61 * Coverage with Repeated Sampling C: 0.87 * Coverage with Repeated Sampling PC: 0.79 * Coverage with Refinement P: 0.63 * Coverage with Refinement C: 1.00 * Coverage with Refinement PC: 0.73 * **Refinement PC:** * Coverage with Direct Generation P: 0.59 * Coverage with Direct Generation C: 0.67 * Coverage with Direct Generation PC: 0.69 * Coverage with Repeated Sampling P: 0.64 * Coverage with Repeated Sampling C: 0.87 * Coverage with Repeated Sampling PC: 0.81 * Coverage with Refinement P: 0.68 * Coverage with Refinement C: 0.83 * Coverage with Refinement PC: 1.00 ### Key Observations * The diagonal elements are all 1.00, indicating perfect coverage when comparing a method to itself. * Repeated Sampling C generally shows high coverage scores with other methods, particularly with Direct Generation PC (0.91), Direct Generation P (0.89), Direct Generation C (0.89), Repeated Sampling P (0.87), Refinement C (0.87), and Repeated Sampling PC (0.85). * Direct Generation P, C, and PC have relatively lower coverage scores with Repeated Sampling C, Refinement C, and Refinement PC compared to Repeated Sampling P and Refinement P. * The lowest coverage scores are generally observed between Direct Generation methods and Repeated Sampling or Refinement methods. ### Interpretation The heatmap provides insights into the similarity and overlap between the outputs of different text generation methods. Higher coverage scores suggest that the methods produce similar outputs, while lower scores indicate more distinct outputs. The high coverage of Repeated Sampling C with other methods suggests that it might generate outputs that are more representative or inclusive of the outputs from other methods. Conversely, the lower coverage between Direct Generation and other methods might indicate that Direct Generation produces more unique or specialized outputs. The "P", "C", and "PC" suffixes likely refer to different configurations or parameters within each generation method. The differences in coverage scores between these configurations suggest that these parameters have a significant impact on the output of each method. The model name "gpt-03-mini" indicates that these coverage scores are specific to this particular language model. The results might vary with different models or datasets.

## Heatmap: Coverage Comparison of Generation Methods ### Overview This image presents a heatmap visualizing the coverage comparison between different text generation methods. The methods are: Direct Generation (Prompt, Completion, and Principal Component), Repeated Sampling (Prompt, Completion, and Principal Component), and Refinement (Prompt, Completion, and Principal Component). The heatmap displays the correlation or similarity between these methods, with values ranging from 0 to 1, indicated by a color gradient. ### Components/Axes * **X-axis:** Represents the generation 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:** Represents the same generation methods as the X-axis, creating a matrix for pairwise comparison. * **Color Scale (Right Side):** Indicates the "Coverage" value, ranging from approximately 0.0 (blue) to 1.8 (red). The scale is not linear, with a concentration of values between 0.5 and 1.0. * **Cell Values:** Each cell in the heatmap displays a numerical value representing the coverage between the corresponding methods. * **Title (Bottom):** "gpt-03-mini" ### Detailed Analysis The heatmap is a 9x9 matrix. Each cell's color corresponds to the coverage value. The diagonal cells (where a method is compared to itself) all have a value of 1.00, indicating perfect coverage. Here's a breakdown of the coverage values, row by row: * **Direct Generation\_P:** * Direct Generation\_P: 1.00 * Direct Generation\_C: 0.74 * Direct Generation\_PC: 0.75 * Repeated Sampling\_P: 0.80 * Repeated Sampling\_C: 0.89 * Repeated Sampling\_PC: 0.84 * Refinement\_P: 0.74 * Refinement\_C: 0.81 * Refinement\_PC: 0.75 * **Direct Generation\_C:** * Direct Generation\_C: 1.00 * Direct Generation\_P: 0.61 * Direct Generation\_PC: 0.71 * Repeated Sampling\_P: 0.68 * Repeated Sampling\_C: 0.89 * Repeated Sampling\_PC: 0.83 * Refinement\_P: 0.76 * Refinement\_C: 0.80 * Refinement\_PC: 0.70 * **Direct Generation\_PC:** * Direct Generation\_PC: 1.00 * Direct Generation\_P: 0.69 * Direct Generation\_C: 0.79 * Repeated Sampling\_P: 0.73 * Repeated Sampling\_C: 0.91 * Repeated Sampling\_PC: 0.84 * Refinement\_P: 0.73 * Refinement\_C: 0.81 * Refinement\_PC: 0.80 * **Repeated Sampling\_P:** * Repeated Sampling\_P: 1.00 * Direct Generation\_P: 0.68 * Direct Generation\_C: 0.71 * Direct Generation\_PC: 0.68 * Repeated Sampling\_C: 0.87 * Repeated Sampling\_PC: 0.88 * Refinement\_P: 0.80 * Refinement\_C: 0.75 * Refinement\_PC: 0.69 * **Repeated Sampling\_C:** * Repeated Sampling\_C: 1.00 * Direct Generation\_P: 0.55 * Direct Generation\_C: 0.67 * Direct Generation\_PC: 0.62 * Repeated Sampling\_P: 0.64 * Repeated Sampling\_PC: 0.80 * Refinement\_P: 0.65 * Refinement\_C: 0.78 * Refinement\_PC: 0.69 * **Repeated Sampling\_PC:** * Repeated Sampling\_PC: 1.00 * Direct Generation\_P: 0.55 * Direct Generation\_C: 0.66 * Direct Generation\_PC: 0.61 * Repeated Sampling\_P: 0.68 * Repeated Sampling\_C: 0.85 * Refinement\_P: 0.76 * Refinement\_C: 0.67 * Refinement\_PC: 0.67 * **Refinement\_P:** * Refinement\_P: 1.00 * Direct Generation\_P: 0.61 * Direct Generation\_C: 0.75 * Direct Generation\_PC: 0.66 * Repeated Sampling\_P: 0.77 * Repeated Sampling\_C: 0.85 * Repeated Sampling\_PC: 0.84 * Refinement\_C: 0.75 * Refinement\_PC: 0.71 * **Refinement\_C:** * Refinement\_C: 1.00 * Direct Generation\_P: 0.56 * Direct Generation\_C: 0.67 * Direct Generation\_PC: 0.62 * Repeated Sampling\_P: 0.61 * Repeated Sampling\_C: 0.87 * Repeated Sampling\_PC: 0.79 * Refinement\_P: 0.63 * Refinement\_PC: 0.73 * **Refinement\_PC:** * Refinement\_PC: 1.00 * Direct Generation\_P: 0.59 * Direct Generation\_C: 0.69 * Direct Generation\_PC: 0.64 * Repeated Sampling\_P: 0.87 * Repeated Sampling\_C: 0.81 * Repeated Sampling\_PC: 0.68 * Refinement\_P: 0.83 * Refinement\_PC: 1.00 ### Key Observations * The diagonal elements are all 1.00, as expected. * "Repeated Sampling\_C" and "Refinement\_C" consistently show higher coverage values with other methods compared to their "P" and "PC" counterparts. * "Direct Generation\_P" and "Refinement\_P" generally have lower coverage values with other methods. * The highest off-diagonal value is 0.91, between "Direct Generation\_PC" and "Repeated Sampling\_C". * The lowest off-diagonal value is 0.55, appearing twice between "Repeated Sampling\_C" and "Direct Generation\_P", and between "Repeated Sampling\_PC" and "Direct Generation\_P". ### Interpretation This heatmap demonstrates the degree of overlap or similarity in the outputs generated by different text generation methods. The "Coverage" metric likely represents how much of the output from one method is also present in the output of another. The higher coverage values between methods ending in "\_C" (Completion) suggest that the completion-based approaches are more consistent and share more common ground with other methods. Conversely, the lower coverage values associated with methods ending in "\_P" (Prompt) indicate that prompt-based approaches may be more diverse or generate more unique outputs. The relatively low coverage between "Direct Generation\_P" and the "Repeated Sampling" and "Refinement" methods suggests that direct prompting leads to significantly different results compared to iterative sampling or refinement techniques. This could be due to the prompt's sensitivity or the inherent randomness in the sampling/refinement processes. The "gpt-03-mini" label suggests this data is specific to a particular language model. The heatmap provides valuable insights into the characteristics of different generation strategies within this model, which can inform the selection of the most appropriate method for a given task. The data suggests that for tasks requiring consistency and overlap with other generation methods, completion-based approaches are preferable. For tasks prioritizing diversity and uniqueness, prompt-based approaches may be more suitable.

## Heatmap: Coverage Correlation Matrix for gpt-o3-mini ### Overview The image is a 9x9 heatmap displaying numerical "Coverage" values between different methodological approaches. The chart is titled "gpt-o3-mini" at the bottom center. The data is presented as a symmetric matrix where each cell's color intensity (from light orange to dark red) corresponds to a coverage value between 0.0 and 1.0, as indicated by a vertical color bar legend on the right side. ### Components/Axes * **Chart Type:** Heatmap (Correlation/Similarity Matrix). * **Title/Label:** "gpt-o3-mini" (located at the bottom center of the chart). * **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: 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 * **Legend (Right Side):** A vertical color bar labeled "Coverage". The scale runs from 0.0 (lightest orange/cream) at the bottom to 1.0 (darkest red) at the top, with a midpoint marker at 0.5. * **Data Cells:** Each cell in the 9x9 grid contains a numerical value (to two decimal places) representing the coverage between the row method and the column method. The diagonal cells (where row and column are identical) all have a value of 1.00 and are the darkest red. ### Detailed Analysis The matrix is symmetric (e.g., value at [Row: Direct Generation C, Column: Direct Generation P] is 0.61, and at [Row: Direct Generation P, Column: Direct Generation C] is 0.74). The following table reconstructs the full data matrix. Values are read directly from the cells. | Row \ Column | Direct Generation P | Direct Generation C | Direct Generation PC | Repeated Sampling P | Repeated Sampling C | Repeated Sampling PC | Refinement P | Refinement C | Refinement PC | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Direct Generation P** | **1.00** | 0.74 | 0.75 | 0.80 | 0.89 | 0.84 | 0.74 | 0.81 | 0.75 | | **Direct Generation C** | 0.61 | **1.00** | 0.71 | 0.68 | 0.89 | 0.83 | 0.76 | 0.80 | 0.70 | | **Direct Generation PC** | 0.69 | 0.79 | **1.00** | 0.73 | 0.91 | 0.84 | 0.73 | 0.81 | 0.80 | | **Repeated Sampling P** | 0.68 | 0.71 | 0.68 | **1.00** | 0.87 | 0.88 | 0.80 | 0.75 | 0.69 | | **Repeated Sampling C** | 0.55 | 0.67 | 0.62 | 0.64 | **1.00** | 0.80 | 0.65 | 0.78 | 0.69 | | **Repeated Sampling PC** | 0.55 | 0.66 | 0.61 | 0.68 | 0.85 | **1.00** | 0.67 | 0.76 | 0.67 | | **Refinement P** | 0.61 | 0.75 | 0.66 | 0.77 | 0.85 | 0.84 | **1.00** | 0.75 | 0.71 | | **Refinement C** | 0.56 | 0.67 | 0.62 | 0.61 | 0.87 | 0.79 | 0.63 | **1.00** | 0.73 | | **Refinement PC** | 0.59 | 0.67 | 0.69 | 0.64 | 0.87 | 0.81 | 0.68 | 0.83 | **1.00** | ### Key Observations 1. **Diagonal Perfection:** All diagonal cells (self-comparison) have a coverage of 1.00, forming a dark red line from top-left to bottom-right. 2. **Highest Off-Diagonal Values:** The highest coverage values outside the diagonal are found in the "Repeated Sampling C" column, particularly when intersected with "Direct Generation PC" (0.91) and "Direct Generation P" (0.89). This suggests strong coverage overlap between these specific method pairs. 3. **Lowest Values:** The lowest coverage values (0.55) occur in the "Repeated Sampling C" and "Repeated Sampling PC" rows when compared against "Direct Generation P". This indicates the weakest coverage relationship in the matrix. 4. **Method Grouping Patterns:** * Methods within the same family (e.g., all "Direct Generation" variants) generally show moderate to high coverage with each other (values typically >0.70). * The "Repeated Sampling" methods show particularly high internal coverage (e.g., Repeated Sampling C vs. Repeated Sampling PC = 0.80). * Coverage between "Refinement" methods and "Direct Generation" methods tends to be lower (often in the 0.55-0.75 range) compared to coverage within the "Repeated Sampling" family. 5. **Color-Value Consistency:** The color gradient accurately reflects the numerical values. Cells with values near 1.0 are dark red, values near 0.5 are medium orange, and values near 0.0 would be light cream (though no values below 0.55 are present). ### Interpretation This heatmap quantifies the "Coverage" relationship between nine different methodological approaches for the `gpt-o3-mini` model. "Coverage" likely measures the degree of overlap, similarity, or agreement in the outputs or capabilities of these methods. * **What the data suggests:** The matrix reveals a hierarchy of methodological similarity. The "Repeated Sampling" family (especially the 'C' variant) appears to be a central, highly interconnected hub, showing strong coverage with most other methods. "Direct Generation" methods form another cluster. "Refinement" methods, while internally consistent, show weaker coverage links to the "Direct Generation" approaches. * **How elements relate:** The symmetric nature confirms that the coverage metric is mutual. The high values between "Repeated Sampling C" and "Direct Generation PC" (0.91) are a key finding, suggesting these two distinct approaches yield highly overlapping results. Conversely, the low values between "Repeated Sampling" methods and "Direct Generation P" (0.55) highlight a significant divergence in what these methods cover. * **Notable anomalies/trends:** The most striking trend is the central role of "Repeated Sampling C," which has the highest average coverage with other methods. An anomaly is the relatively low coverage (0.61) between the two most basic methods: "Direct Generation C" and "Direct Generation P," suggesting their outputs are quite distinct despite sharing a common high-level approach. The "PC" variants (likely meaning "Prompt + Chain-of-thought" or similar) generally show higher coverage with other methods than their "P" or "C" counterparts alone, indicating a more comprehensive or overlapping coverage profile.

# Technical Document: Coverage Comparison Across Methods and Models ## Image Description The image is a **heatmap** visualizing coverage values across different **methods** (rows) and **models** (columns). The color intensity corresponds to coverage values, with darker red indicating higher coverage (closer to 1.0) and lighter red indicating lower coverage (closer to 0.0). The legend on the right maps colors to numerical values. --- ## Key Components ### 1. **Axis Labels** - **X-axis (Models)**: `gpt-o3-mini`, `gpt-o3-medium`, `gpt-o3-large` - **Y-axis (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` - **Title**: `Coverage Comparison Across Methods and Models` ### 2. **Legend** - **Location**: Right side of the heatmap. - **Color Scale**: - `0.0` (lightest red) to `1.0` (darkest red). - Intermediate values: `0.5`, `0.75`, `0.9`. --- ## Data Structure The heatmap is a **9x3 matrix** (9 methods × 3 models). Each cell contains a numerical coverage value. Below is the reconstructed table: | Method | gpt-o3-mini | gpt-o3-medium | gpt-o3-large | |----------------------|-------------|---------------|--------------| | Direct Generation P | 1.00 | 0.74 | 0.75 | | Direct Generation C | 0.61 | 1.00 | 0.71 | | Direct Generation PC | 0.69 | 0.79 | 1.00 | | Repeated Sampling P | 0.68 | 0.71 | 0.68 | | Repeated Sampling C | 0.55 | 0.67 | 0.62 | | Repeated Sampling PC | 0.55 | 0.66 | 0.61 | | Refinement P | 0.61 | 0.75 | 0.66 | | Refinement C | 0.56 | 0.67 | 0.62 | | Refinement PC | 0.59 | 0.67 | 0.69 | --- ## Trends and Observations 1. **Direct Generation P** consistently shows high coverage (1.00 for `gpt-o3-mini`, 0.74–0.89 for other models). 2. **Direct Generation C** and **PC** exhibit moderate coverage, with PC achieving 1.00 for `gpt-o3-large`. 3. **Repeated Sampling** methods generally have lower coverage (0.55–0.88), with `Repeated Sampling C` showing the lowest values. 4. **Refinement** methods demonstrate moderate-to-high coverage, with `Refinement PC` achieving 0.83 for `gpt-o3-large`. 5. **Color Consistency**: - Dark red cells (e.g., 1.00) align with the legend's highest value. - Lighter red cells (e.g., 0.55–0.62) match the legend's lower range. --- ## Spatial Grounding - **Legend Position**: Right-aligned, vertically stacked. - **X-axis Position**: Bottom of the heatmap. - **Y-axis Position**: Left of the heatmap. --- ## Notes - All values are in **English** and represent coverage percentages. - No non-English text is present. - The heatmap uses a **continuous color scale** without discrete categories. This analysis confirms that the heatmap effectively compares coverage across methods and models, with `Direct Generation P` and `Refinement PC` showing the highest performance for `gpt-o3-large`.