micro_cartan_attn - Cartan Matrix Attention Mechanisms

Cartan matrix-constrained attention mechanisms for semantic coherence

This crate implements attention mechanisms that maintain Cartan matrix constraints for preserving orthogonal semantic relationships in neural networks. It provides structured attention patterns based on Lie algebra principles.

✅ Implemented Features

• CartanAttention: Attention mechanism enforcing Cartan matrix constraints
• MultiHeadCartanAttention: Multi-head attention with different Cartan matrix types
• Compliance-based Scoring: Attention weights based on Cartan constraint adherence
• Orthogonalization: Basic Gram-Schmidt orthogonalization (placeholder)
• Regularization: Constraint enforcement during training (placeholder)

❌ Not Yet Implemented

• Complete Orthogonalizer: Only stub implementation exists
• Training Integration: No actual optimization loops
• Complex Cartan Types: Only identity and basic types implemented
• SIMD Attention: No vectorized attention computations
• Positional Encoding: Placeholder implementation only

📦 Installation

Add this to your Cargo.toml:

[dependencies]
micro_cartan_attn = { path = "../micro_cartan_attn" }
micro_core = { path = "../micro_core" }

🏗️ Core Components

CartanAttention

Basic Cartan-constrained attention mechanism:

use micro_cartan_attn::{CartanAttention, AttentionConfig};
use micro_core::{RootVector, CartanMatrix};

// Create Cartan matrix (identity by default)
let cartan_matrix = CartanMatrix::default();

// Configure attention
let config = AttentionConfig {
    orthogonalize_output: true,
    regularize_attention: true,
    temperature: 1.0,
    normalize_input: true,
};

// Create attention mechanism
let mut attention = CartanAttention::with_config(cartan_matrix, config)?;

// Apply attention to vector sequence
let vectors = vec![
    RootVector::from_array([1.0, 0.0, /* ... 30 more zeros ... */]),
    RootVector::from_array([0.0, 1.0, /* ... 30 more zeros ... */]),
];

let attended = attention.apply_attention(&vectors)?;
let weights = attention.attention_weights()?;

MultiHeadCartanAttention

Multi-head attention with different Cartan constraint types:

use micro_cartan_attn::MultiHeadCartanAttention;

// Create different Cartan matrices for each head
let matrices = vec![
    CartanMatrix::default(),         // Identity
    CartanMatrix::default(),         // Another identity (placeholder)
];

let mut multi_head = MultiHeadCartanAttention::new(matrices)?;

// Set custom output weights
multi_head.set_output_weights(vec![0.6, 0.4])?;

// Apply multi-head attention
let output = multi_head.apply_attention(&input_vectors)?;

🧮 Mathematical Foundation

Cartan Matrix Constraints

The attention mechanism enforces compliance with Cartan matrix relationships:

// Attention scoring based on Cartan compliance
for (i, vector) in vectors.iter().enumerate() {
    let mut score = 0.0;
    
    // Score based on inner product compliance
    for (j, other_vector) in vectors.iter().enumerate() {
        if i != j {
            let actual_inner = vector.dot(other_vector);
            let target_inner = cartan_matrix.entry(i, j);
            let compliance = (-((actual_inner - target_inner).powi(2))).exp();
            score += compliance;
        }
    }
    
    // Score based on norm compliance (Cartan normalization)
    let norm = vector.magnitude();
    let target_norm = (2.0_f32).sqrt(); // ⟨αᵢ, αᵢ⟩ = 2
    let norm_compliance = (-((norm - target_norm).powi(2))).exp();
    score += norm_compliance;
}

Attention Process

1. Input Normalization: Optional vector normalization
2. Compliance Scoring: Compute how well vectors satisfy Cartan constraints
3. Softmax Weighting: Apply temperature scaling and softmax normalization
4. Attention Application: Weight vectors by attention scores
5. Regularization: Apply constraint enforcement (placeholder)
6. Orthogonalization: Maintain orthogonal relationships (placeholder)

🔧 Configuration

AttentionConfig

use micro_cartan_attn::AttentionConfig;

let config = AttentionConfig {
    orthogonalize_output: true,    // Apply orthogonalization after attention
    regularize_attention: true,    // Apply regularization during attention
    temperature: 1.0,              // Temperature for softmax
    normalize_input: true,         // Normalize input vectors
};

Feature Flags

[features]
default = ["gram-schmidt"]
std = ["serde?/std", "nalgebra/std"]
serde = ["dep:serde", "dep:serde_json"]
gram-schmidt = []               # Gram-Schmidt orthogonalization (placeholder)
training = []                   # Training-time features (placeholder)
simd = []                      # SIMD optimizations (not implemented)
analysis = []                  # Analysis tools (not implemented)

📊 Performance

Current Implementation

The current implementation focuses on correctness over performance:

• Attention Computation: O(n²) for n vectors
• Memory Usage: Minimal allocation, works in no_std
• SIMD: Not yet implemented
• Batching: Not yet implemented

Benchmarks (Estimated)

⚠️ Current Limitations

1. Stub Implementations: Many components are placeholders
2. No Training Integration: Missing optimization and backpropagation
3. Limited Cartan Types: Only identity matrices fully implemented
4. No SIMD: Performance not optimized
5. Basic Testing: Limited test coverage
6. Missing Features: Positional encoding, masking, etc.

🧪 Testing

# Run unit tests
cargo test

# Test with micro_core integration
cargo test --features integration-tests

# Test serialization
cargo test --features serde

Test Coverage

Current tests cover:

• Basic attention mechanism creation
• Attention score computation
• Multi-head attention functionality
• Integration with micro_core types

Missing tests:

• Property-based testing for mathematical correctness
• Performance benchmarking
• Edge case handling
• Cartan constraint verification

📈 Roadmap

Phase 1: Core Functionality

• Complete orthogonalization implementations
• Add more Cartan matrix types (A_n, D_n, E_8)
• Implement proper regularization
• Add comprehensive testing

Phase 2: Performance

• SIMD-accelerated attention computation
• Batch processing support
• Memory-efficient attention for long sequences
• Performance benchmarking suite

Phase 3: Advanced Features

• Training integration with backpropagation
• Positional encoding implementations
• Attention masking support
• Analysis and visualization tools

📚 Examples

Current examples demonstrate:

• Basic attention mechanism usage
• Multi-head attention configuration
• Integration with micro_core types

Planned examples:

• Training integration examples
• Performance optimization examples
• Cartan constraint analysis

🤝 Contributing

This is part of a research project. Contributions welcome for:

• Implementing placeholder components
• Adding comprehensive tests
• Performance optimizations
• Mathematical correctness verification
• Documentation improvements

📄 License

Licensed under either of:

• Apache License, Version 2.0 (LICENSE-APACHE)
• MIT License (LICENSE-MIT)

at your option.

🔗 Related Crates

• micro_core: Core types and mathematical structures
• micro_routing: Dynamic routing using attention mechanisms
• micro_metrics: Performance monitoring and analysis
• micro_swarm: High-level orchestration with attention-based coordination

Part of the rUv-FANN Semantic Cartan Matrix system - Research implementation of Cartan matrix-inspired attention mechanisms.