# Technical Document Extraction: Heatmap Analysis of $\phi(A_{r=64}, A_{r=8}, i, j)$
## 1. Header Information
* **Main Title (Mathematical Expression):** $\phi(A_{r=64}, A_{r=8}, i, j)$
* **Context:** This image appears to be a visualization of subspace similarity or projection weights between two Low-Rank Adaptation (LoRA) matrices of different ranks ($r=64$ and $r=8$).
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## 2. Component Isolation
The image is segmented into four distinct heatmap panels arranged horizontally, sharing a common vertical axis and a color scale.
### A. Global Legend (Color Bar)
* **Location:** Far right of the image.
* **Type:** Continuous gradient scale.
* **Range:** $0.0$ to $1.0$.
* **Color Mapping:**
* **0.0 (Dark Purple/Black):** Low correlation/value.
* **0.5 (Magenta/Red):** Mid-range value.
* **1.0 (Light Peach/White):** High correlation/value.
* **Background Color:** The light grey/lavender areas in the plots represent null or zero values outside the defined matrix dimensions.
### B. Shared Vertical Axis (y-axis)
* **Label:** $i$
* **Markers:** $1, 2, 3, 4, 5, 6, 7, 8$ (Top to Bottom).
* **Description:** Represents the row index, likely corresponding to the rank dimensions of the $r=8$ matrix.
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## 3. Detailed Panel Analysis
### Panel 1: $\Delta W_q$ (Wide)
* **Title:** $\Delta W_q$
* **Horizontal Axis ($j$):** Range $1$ to $64$. Markers shown: $1, 6, 12, 18, 23, 29, 35, 40, 46, 52, 58$.
* **Visual Trend:** A "staircase" or triangular pattern.
* For each row $i$, the values are high (bright magenta, approx. $0.4 - 0.6$) starting from a specific $j$ index and continuing to the right.
* As $i$ increases, the starting point of the colored block shifts to the right (higher $j$).
* The top row ($i=1$) is colored across almost the entire width.
* The bottom row ($i=8$) only begins its colored section around $j=6$.
### Panel 2: $\Delta W_v$ (Wide)
* **Title:** $\Delta W_v$
* **Horizontal Axis ($j$):** Range $1$ to $64$. Markers identical to Panel 1.
* **Visual Trend:** Nearly identical to Panel 1. It shows the same staggered, lower-triangular-like distribution of weights across the 64 dimensions of the larger rank matrix.
### Panel 3: $\Delta W_q$ (Square)
* **Title:** $\Delta W_q$
* **Horizontal Axis ($j$):** Range $1$ to $8$. Markers: $1, 2, 3, 4, 5, 6, 7, 8$.
* **Visual Trend:** A clear **Lower Triangular Matrix**.
* **Diagonal:** Values are highest along the diagonal where $i=j$ (darker magenta, approx $0.3 - 0.4$).
* **Column 1:** The first column ($j=1$) has a consistently higher value (brightest magenta, approx $0.5$) across all rows $i=1$ to $8$.
* **Upper Triangle:** The area where $j > i$ is light grey (null/zero).
* **Lower Triangle:** The area where $i \ge j$ contains data, with values generally decreasing (getting darker) as $i$ and $j$ increase, except for the prominent first column.
### Panel 4: $\Delta W_v$ (Square)
* **Title:** $\Delta W_v$
* **Horizontal Axis ($j$):** Range $1$ to $8$. Markers identical to Panel 3.
* **Visual Trend:** Identical structure to Panel 3.
* Shows a lower triangular pattern.
* Strongest values (approx $0.5$) are in the first column ($j=1$).
* The diagonal and lower sub-diagonal elements are present but show lower intensity (darker purple) compared to the first column.
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## 4. Data Summary Table (Representative Values)
| Feature | Panel 1 & 2 ($j=1 \dots 64$) | Panel 3 & 4 ($j=1 \dots 8$) |
| :--- | :--- | :--- |
| **Pattern** | Staggered Horizontal Blocks | Lower Triangular |
| **Max Value Approx.** | $0.5 - 0.6$ | $0.5$ (at $j=1$) |
| **Min Value (Data)** | $\approx 0.1$ | $\approx 0.1$ |
| **Null Value** | Background Grey ($0.0$) | Background Grey ($0.0$) |
## 5. Technical Conclusion
The visualization demonstrates the relationship between the subspaces of two different rank adaptations. The "Wide" plots show how the 8 dimensions of the smaller rank relate to the 64 dimensions of the larger rank. The "Square" plots show the internal relationship or projection within the first 8 dimensions. The consistent lower-triangular shape suggests an ordered dependency or a Gram-Schmidt-like orthogonalization where lower indices capture broader information (indicated by the high-intensity first column).
