# Technical Document Extraction: Heatmap Analysis
## **Key Components**
### **Panels**
- **Panel 1 (Top-Left):** ΔW_q (Layer i vs. j)
- **Panel 2 (Top-Right):** ΔW_v (Layer i vs. j)
- **Panel 3 (Bottom-Left):** φ(A_r=8, A_r=64, i, j) (Layer i vs. j)
- **Panel 4 (Bottom-Right):** φ(A_r=8, A_r=64, i, j) (Layer i vs. j)
### **Axes**
- **X-Axis (Horizontal):** `j` (Columns), labeled 1 to 8
- **Y-Axis (Vertical):** `Layer i` (Rows), labeled 1 to 8
### **Legend**
- **Color Scale:**
- **Range:** 0.0 (dark purple) to 1.0 (light orange)
- **Gradient:** Linear transition from dark purple (low values) to light orange (high values)
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## **Data Structure**
### **Categories**
- **Layers (i):** 1, 2, 3, 4, 5, 6, 7, 8
- **Columns (j):** 1, 2, 3, 4, 5, 6, 7, 8
### **Heatmap Values**
- **ΔW_q (Top Panels):**
- **Layer 1:** Values range from ~0.8 (j=1) to ~0.2 (j=8)
- **Layer 8:** Values range from ~0.6 (j=1) to ~0.0 (j=8)
- **Trend:** Decreasing intensity from top-left (high values) to bottom-right (low values).
- **ΔW_v (Top Panels):**
- **Layer 1:** Values range from ~0.8 (j=1) to ~0.4 (j=8)
- **Layer 8:** Values range from ~0.6 (j=1) to ~0.0 (j=8)
- **Trend:** Similar to ΔW_q but with slightly higher values in upper layers.
- **φ(A_r=8, A_r=64, i, j) (Bottom Panels):**
- **Layer 1:** Values range from ~0.8 (j=1) to ~0.4 (j=8)
- **Layer 8:** Values range from ~0.6 (j=1) to ~0.0 (j=8)
- **Trend:** Triangular pattern (dark purple in lower-left, light orange in upper-right).
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## **Key Observations**
1. **ΔW_q and ΔW_v:**
- Higher values (lighter colors) dominate in **Layer 1** across all `j`.
- Values decay exponentially with increasing `Layer i` and `j`.
2. **φ(A_r=8, A_r=64, i, j):**
- **Triangular Pattern:**
- **High values (light orange)** concentrated in **upper-right** (small `i`, large `j`).
- **Low values (dark purple)** concentrated in **lower-left** (large `i`, small `j`).
- Suggests a correlation between `Layer i` and `j` indices.
3. **Color Consistency:**
- All panels share the same color scale (0.0–1.0), enabling direct comparison.
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## **Technical Notes**
- **Heatmap Design:**
- **Triangular Masking:** Panels 3 and 4 (φ) use a triangular mask, truncating values where `i > j`.
- **Normalization:** Values are normalized to the 0.0–1.0 scale for consistency.
- **Interpretation:**
- ΔW_q and ΔW_v represent weight differences (e.g., between layers or models).
- φ likely represents a similarity or correlation metric between layers `i` and `j` for two architectures (`A_r=8` and `A_r=64`).
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## **Data Table Reconstruction**
| Layer i | j=1 | j=2 | j=3 | j=4 | j=5 | j=6 | j=7 | j=8 |
|---------|-----|-----|-----|-----|-----|-----|-----|-----|
| **ΔW_q** | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 |
| **ΔW_v** | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 |
| **φ(A_r=8)** | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 |
| **φ(A_r=64)** | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0.0 | 0.0 |
*Note: Values are approximate based on color intensity.*
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## **Conclusion**
The heatmaps reveal layer-wise dependencies and correlations in neural network architectures. ΔW_q and ΔW_v show diminishing differences with depth, while φ highlights architectural similarities between layers. The triangular pattern in φ suggests a hierarchical relationship between `i` and `j`.
