# Tropical Algebra GPU Integration Plan for v0.9.3
## Overview
This document outlines the GPU acceleration strategy for amari-tropical operations, building on the existing WebGPU/wgpu infrastructure in amari-gpu.
## Current GPU Infrastructure Analysis
### Existing Capabilities
- ✅ WebGPU/wgpu setup with adaptive GPU/CPU dispatch
- ✅ Compute shader pipeline framework
- ✅ Buffer management and staging
- ✅ Batch processing optimizations
- ✅ Geometric algebra operations (geometric product, batch operations)
- ✅ Information geometry operations (Amari-Chentsov tensor, Fisher matrices)
### Integration Points
- `GpuCliffordAlgebra` - Base GPU compute infrastructure
- `AdaptiveCompute` - Intelligent CPU/GPU dispatch
- Compute shader framework with WGSL
- Buffer management with bytemuck serialization
## Tropical Operations for GPU Acceleration
### High Priority (Immediate GPU Benefits)
#### 1. Tropical Matrix Multiplication
**Current State**: Basic WASM implementation exists
**GPU Benefits**: Massive parallelization for large matrices
**Implementation**:
- Extend existing geometric product infrastructure
- WGSL compute shader for tropical matrix ops
- Batch processing for multiple matrix operations
```rust
// Target API
pub async fn tropical_matrix_multiply_batch(
&self,
a_matrices: &[TropicalMatrix],
b_matrices: &[TropicalMatrix],
) -> Result<Vec<TropicalMatrix>, GpuError>
```
#### 2. Viterbi Algorithm Acceleration
**Current State**: CPU-only implementation in amari-tropical
**GPU Benefits**: Parallel state computation, large sequence processing
**Implementation**:
- GPU-optimized forward/backward passes
- Parallel Viterbi trellis computation
- Batch decoding for multiple sequences
```rust
// Target API
pub async fn viterbi_decode_batch(
&self,
hmms: &[TropicalViterbi],
observation_sequences: &[Vec<usize>],
) -> Result<Vec<(Vec<usize>, TropicalNumber)>, GpuError>
```
#### 3. All-Pairs Shortest Paths
**Current State**: Basic implementation exists
**GPU Benefits**: Parallel Floyd-Warshall, large graph processing
**Implementation**:
- GPU Floyd-Warshall with work-efficient parallelization
- Blocked algorithm for memory efficiency
- Batch processing for multiple graphs
### Medium Priority (Optimization Algorithms)
#### 4. Tropical Convolution
**Use Case**: Sequence analysis, signal processing in tropical algebra
**GPU Benefits**: Parallel convolution computation
**Implementation**: WGSL compute shaders for tropical convolution
#### 5. Tropical Polynomial Operations
**Use Case**: Optimization, root finding
**GPU Benefits**: Parallel evaluation, derivative computation
**Implementation**: Vectorized polynomial evaluation and operations
### Lower Priority (Advanced Operations)
#### 6. Tropical Eigenvalue Computation
**Use Case**: Advanced tropical linear algebra
**GPU Benefits**: Iterative algorithms, large matrix eigenanalysis
## Implementation Architecture
### 1. Core GPU Tropical Infrastructure
```rust
/// GPU-accelerated tropical algebra operations
pub struct GpuTropicalAlgebra {
device: wgpu::Device,
queue: wgpu::Queue,
// Compute pipelines for different operations
matrix_multiply_pipeline: wgpu::ComputePipeline,
viterbi_pipeline: wgpu::ComputePipeline,
shortest_path_pipeline: wgpu::ComputePipeline,
convolution_pipeline: wgpu::ComputePipeline,
}
impl GpuTropicalAlgebra {
pub async fn new() -> Result<Self, GpuError>;
// Matrix operations
pub async fn tropical_matrix_multiply_batch(...) -> Result<...>;
pub async fn tropical_matrix_power_batch(...) -> Result<...>;
// HMM operations
pub async fn viterbi_decode_batch(...) -> Result<...>;
pub async fn forward_backward_batch(...) -> Result<...>;
// Graph operations
pub async fn all_pairs_shortest_paths_batch(...) -> Result<...>;
pub async fn tropical_closure_batch(...) -> Result<...>;
// Signal processing
pub async fn tropical_convolution_batch(...) -> Result<...>;
pub async fn tropical_filter_batch(...) -> Result<...>;
}
```
### 2. Integration with Existing amari-gpu
```rust
// Extend AdaptiveCompute to include tropical operations
impl AdaptiveCompute {
pub async fn new_with_tropical<const P: usize, const Q: usize, const R: usize>() -> Self {
let gpu = GpuCliffordAlgebra::new::<P, Q, R>().await.ok();
let tropical_gpu = GpuTropicalAlgebra::new().await.ok();
Self { gpu, tropical_gpu }
}
}
```
### 3. WGSL Compute Shaders
#### Tropical Matrix Multiplication
```wgsl
@group(0) @binding(0)
var<storage, read> matrix_a: array<f32>;
@group(0) @binding(1)
var<storage, read> matrix_b: array<f32>;
@group(0) @binding(2)
var<storage, read_write> result: array<f32>;
@compute @workgroup_size(16, 16)
fn tropical_matrix_multiply(
@builtin(global_invocation_id) global_id: vec3<u32>
) {
let row = global_id.x;
let col = global_id.y;
if (row >= rows_a || col >= cols_b) {
return;
}
var max_val = -1e30; // Tropical zero (negative infinity)
for (var k = 0u; k < cols_a; k++) {
let a_val = matrix_a[row * cols_a + k];
let b_val = matrix_b[k * cols_b + col];
// Tropical multiplication: addition
// Tropical addition: maximum
max_val = max(max_val, a_val + b_val);
}
result[row * cols_b + col] = max_val;
}
```
#### Viterbi Algorithm
```wgsl
@group(0) @binding(0)
var<storage, read> transition_matrix: array<f32>;
@group(0) @binding(1)
var<storage, read> emission_matrix: array<f32>;
@group(0) @binding(2)
var<storage, read> observations: array<u32>;
@group(0) @binding(3)
var<storage, read_write> trellis: array<f32>;
@compute @workgroup_size(64)
fn viterbi_forward_step(
@builtin(global_invocation_id) global_id: vec3<u32>
) {
let state = global_id.x;
let time = global_id.y;
if (state >= num_states || time >= sequence_length) {
return;
}
if (time == 0u) {
// Initialization
trellis[state] = emission_matrix[state * num_observations + observations[0]];
return;
}
var max_score = -1e30;
// Find best previous state
for (var prev_state = 0u; prev_state < num_states; prev_state++) {
let prev_score = trellis[(time - 1u) * num_states + prev_state];
let transition_score = transition_matrix[prev_state * num_states + state];
let emission_score = emission_matrix[state * num_observations + observations[time]];
let total_score = prev_score + transition_score + emission_score;
max_score = max(max_score, total_score);
}
trellis[time * num_states + state] = max_score;
}
```
### 4. Adaptive Dispatch Strategy
```rust
impl GpuTropicalAlgebra {
/// Determine if GPU acceleration should be used
pub fn should_use_gpu_for_operation(operation: TropicalOperation, size: usize) -> bool {
match operation {
TropicalOperation::MatrixMultiply => size >= 64, // 64x64 matrices or larger
TropicalOperation::ViterbiDecode => size >= 1000, // Sequences >= 1000 steps
TropicalOperation::ShortestPaths => size >= 100, // Graphs >= 100 nodes
TropicalOperation::Convolution => size >= 512, // Convolution >= 512 points
}
}
}
```
## Performance Targets
### Benchmarking Goals
- **Matrix Multiplication**: 10-100x speedup for 1000x1000+ matrices
- **Viterbi Decoding**: 5-50x speedup for sequences > 1000 steps
- **Shortest Paths**: 10-100x speedup for graphs > 100 nodes
- **Memory Efficiency**: < 2x GPU memory overhead vs optimal
### Memory Optimization
- Efficient buffer reuse and pooling
- Staging buffer management for large datasets
- Streaming computation for memory-constrained operations
## Integration Timeline
### Phase 1: Core Infrastructure (Week 1)
- [ ] Extend amari-gpu with tropical algebra support
- [ ] Implement `GpuTropicalAlgebra` base structure
- [ ] Create tropical matrix multiplication GPU pipeline
- [ ] Add adaptive dispatch for tropical operations
### Phase 2: HMM Acceleration (Week 2)
- [ ] Implement GPU Viterbi algorithm
- [ ] Add forward-backward algorithm GPU support
- [ ] Batch processing for multiple HMM operations
- [ ] Integration with existing amari-tropical types
### Phase 3: Graph Algorithms (Week 3)
- [ ] GPU all-pairs shortest paths
- [ ] Tropical matrix powers and closure
- [ ] Graph analysis batch operations
- [ ] Performance optimization and profiling
### Phase 4: Advanced Operations (Week 4)
- [ ] Tropical convolution and filtering
- [ ] Polynomial operations GPU acceleration
- [ ] Advanced linear algebra operations
- [ ] Comprehensive benchmarking and optimization
## Testing Strategy
### Unit Tests
- GPU vs CPU correctness verification
- Edge case handling (empty matrices, degenerate cases)
- Memory safety and buffer management
- Adaptive dispatch correctness
### Performance Tests
- Scaling benchmarks for different operation sizes
- Memory usage profiling
- Comparative analysis vs CPU implementations
- Real-world workload simulation
### Integration Tests
- End-to-end tropical algebra workflows
- Cross-platform compatibility (different GPU vendors)
- Fallback behavior when GPU unavailable
- WASM integration compatibility
## Risk Mitigation
### Technical Risks
1. **GPU Memory Limitations**: Implement streaming algorithms and chunked processing
2. **Precision Issues**: Validate f32 vs f64 precision requirements
3. **Platform Compatibility**: Extensive testing across GPU vendors
4. **WebGPU Limitations**: Design fallbacks for unsupported features
### Performance Risks
1. **Small Operation Overhead**: Careful threshold tuning for GPU dispatch
2. **Memory Transfer Costs**: Minimize CPU-GPU data movement
3. **Synchronization Overhead**: Asynchronous operation design
## Success Metrics
### Quantitative Goals
- [ ] 10x+ speedup for large tropical matrix operations
- [ ] 5x+ speedup for large Viterbi decoding
- [ ] <10% CPU memory overhead for GPU operations
- [ ] 95%+ test coverage for GPU code paths
### Qualitative Goals
- [ ] Seamless integration with existing amari-tropical API
- [ ] Minimal breaking changes to existing code
- [ ] Clear performance characteristics documentation
- [ ] Robust error handling and graceful degradation
## Future Extensions
### WebGPU in WASM
- Extend tropical GPU operations to run in WebAssembly
- Browser-based tropical algebra acceleration
- Integration with existing WASM bindings
### Advanced Algorithms
- Tropical neural networks on GPU
- Large-scale optimization algorithms
- Distributed tropical computation
### Machine Learning Integration
- GPU-accelerated tropical SVMs
- Tropical deep learning primitives
- High-performance tropical feature extraction
---
This plan provides a comprehensive roadmap for integrating tropical algebra operations with GPU acceleration, leveraging the existing amari-gpu infrastructure while providing substantial performance improvements for computationally intensive tropical operations.