# Technical Document Extraction ## Heatmap Analysis (Top Section) ### MLP (-5.3k Params) - **Label**: "MLP (-5.3k Params) Fail at Heavy Tails" - **Visual Characteristics**: - Dominant orange gradient with a central dark red core - Color scale: Logarithmic (10⁻¹⁰ to 10⁻²) - Spatial coordinates: X-axis (Spatial Coordinate), Y-axis (Potential) ### KAN (-5.2k Params) - **Label**: "KAN (-5.2k Params) Spline Oscillations" - **Visual Characteristics**: - Grid-like pattern with alternating bright/dark regions - Color scale: Same logarithmic scale as MLP - Notable: Blue boundary outline around the heatmap ### RAN (Ours, 72 Params) - **Label**: "RAN (Ours, 72 Params) Structural Resonance" - **Visual Characteristics**: - Uniform dark blue background - Green boundary outline - Color scale: Same logarithmic scale ## Cross-Section Analysis (Bottom Left) ### Axes - **X-axis**: Spatial Coordinate (x=0) - **Y-axis**: Potential (f(x)) ### Data Series 1. **Ground Truth** (Gray dashed line) - Smooth U-shaped curve - Peak at x=0, y=1.0 2. **MLP (Tail Bias)** (Red dashed line) - Tail bias visible at x=±3 - Peak shifted downward (y≈0.8) 3. **KAN (Ripple)** (Blue dotted line) - Oscillations near x=±1 - Peak at x=0, y=0.9 4. **RAN (Ours)** (Green solid line) - Smoothest curve - Peak at x=0, y=1.0 ### Inset Zoom - **Region**: x ∈ [-1, 1], y ∈ [0.2, 1.0] - **Purpose**: Highlight peak fidelity differences ## Efficiency Gap Analysis (Bottom Right) ### Bar Chart - **X-axis**: Model Type (MLP, KAN, RAN) - **Y-axis (Left)**: Parameter Count (Log scale: 10¹ to 10⁴) - MLP: 5,300 parameters - KAN: 5,200 parameters - RAN: 72 parameters - **Y-axis (Right)**: MSE Loss (Log scale: 10⁻¹⁰ to 10⁻¹) - MLP: 10⁻³ - KAN: 10⁻⁴ - RAN: 10⁻¹⁰ ### Annotations - **Text**: "72 params beat 5k+ params (100x lower MSE vs. KAN)" - **Visual**: Arrows connecting RAN bar to MLP/KAN bars ## Legend & Spatial Grounding ### Cross-Section Legend - **Position**: Right side of graph - **Color Mapping**: - Gray: Ground Truth - Red: MLP (Tail Bias) - Blue: KAN (Ripple) - Green: RAN (Ours) ### Bar Chart Legend - **Implicit**: Colors match bar colors (Red=MLP, Blue=KAN, Green=RAN) ## Key Trends 1. **Parameter Efficiency**: - RAN uses 72 parameters (99.9% fewer than MLP/KAN) - Achieves 100x lower MSE than KAN 2. **Topological Performance**: - MLP exhibits tail bias at x=±3 - KAN shows spline oscillations near x=±1 - RAN maintains ground truth fidelity 3. **Resonance Patterns**: - RAN's uniform dark blue suggests structural resonance - KAN's grid pattern indicates spline oscillations