# UHG Compliance Improvements

## 1. UHG ReLU Research and Implementation
- Study mathematical implications of different UHG ReLU approaches:
  - Pure projective approach (current library version)
  - Feature-separated approach (alternative version)
  - Mathematical theory behind both versions
  - Impact on cross-ratio preservation
  - Relationship to UHG principles

- Design and run comparative experiments:
  - Performance benchmarks
  - Stability analysis
  - Cross-ratio preservation metrics
  - Impact on gradient flow
  - Training convergence comparison

- Current alternative version for comparison:
  ```python
  def uhg_relu(x: torch.Tensor, eps: float = 1e-8) -> torch.Tensor:
      # Split features and homogeneous coordinate
      features = x[..., :-1]
      h_coord = x[..., -1:]
      
      # Apply ReLU to features only
      activated_features = F.relu(features)
      
      # Concatenate back with homogeneous coordinate
      out = torch.cat([activated_features, h_coord], dim=-1)
      
      # Normalize to maintain projective structure
      norm = torch.norm(out[..., :-1], p=2, dim=-1, keepdim=True)
      norm = torch.clamp(norm, min=eps)
      
      # Scale features while preserving homogeneous coordinate
      out = torch.cat([
          out[..., :-1] / norm,
          out[..., -1:]
      ], dim=-1)
      
      return out
  ```

- Based on research results:
  - Document mathematical findings
  - Make implementation recommendations
  - Update library implementation if needed
  - Add tests for chosen approach
  - Provide usage guidelines

## 2. Add Cross-Ratio Preservation Monitoring
- Implement cross-ratio evaluation function
- Add training-time monitoring
- Create cross-ratio based regularization
- Add visualization tools for cross-ratio preservation
- Document cross-ratio importance in UHG

## 3. Implement UHG-Compliant Attention
- Create hyperbolic attention mechanism
- Ensure attention weights preserve UHG structure
- Handle homogeneous coordinates in attention computation
- Add tests for attention UHG compliance
- Document attention implementation details

## 4. Improve UHG-Compliant Aggregation
- Update neighbor aggregation to be fully UHG-compliant
- Implement proper hyperbolic mean operation
- Add UHG-aware edge weight handling
- Create tests for aggregation UHG compliance
- Document aggregation mathematical foundations

## 5. Add UHG-Compliant Normalization Layers
- Implement batch normalization that preserves UHG structure
- Add layer normalization for hyperbolic space
- Create feature scaling methods for projective space
- Add tests for normalization UHG compliance
- Document normalization mathematical basis

## 6. Implement UHG-Compliant Initialization
- Create weight initialization in hyperbolic space
- Ensure initial cross-ratio preservation
- Add proper homogeneous coordinate initialization
- Create tests for initialization UHG compliance
- Document initialization best practices

## Priority Order
1. UHG ReLU Research and Implementation (Most urgent as current implementation is not fully compliant)
2. Cross-Ratio Preservation (Fundamental to UHG correctness)
3. Normalization Layers (Important for training stability)
4. Initialization (Important for training start)
5. Aggregation (Important for graph operations)
6. Attention (Can be added after other fundamentals)

## Notes
- Each improvement should include:
  - Mathematical justification from UHG principles
  - Tests for UHG compliance
  - Documentation with examples
  - Performance benchmarks
  - Visualization tools where applicable 