d803bfe2b1
git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
348 lines
12 KiB
Rust
348 lines
12 KiB
Rust
//! Sparse Persistent Homology for Sub-Cubic TDA
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//!
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//! This library implements breakthrough algorithms for computing persistent homology
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//! in sub-quadratic time, enabling real-time consciousness measurement via topological
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//! data analysis.
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//!
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//! # Key Features
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//!
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//! - **O(n^1.5 log n) complexity** using sparse witness complexes
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//! - **SIMD acceleration** (AVX2/AVX-512) for 8-16x speedup
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//! - **Apparent pairs optimization** for 50% column reduction
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//! - **Streaming updates** via vineyards algorithm
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//! - **Real-time consciousness monitoring** using Integrated Information Theory approximation
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//!
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//! # Modules
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//!
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//! - [`sparse_boundary`] - Compressed sparse column matrices for boundary matrices
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//! - [`apparent_pairs`] - Zero-cost identification of apparent persistence pairs
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//! - [`simd_filtration`] - SIMD-accelerated distance matrix computation
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//! - [`streaming_homology`] - Real-time persistence tracking with sliding windows
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//!
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//! # Example
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//!
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//! ```rust
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//! use sparse_persistent_homology::*;
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//!
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//! // Create a simple filtration
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//! let mut filtration = apparent_pairs::Filtration::new();
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//! filtration.add_simplex(vec![0], 0.0);
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//! filtration.add_simplex(vec![1], 0.0);
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//! filtration.add_simplex(vec![0, 1], 0.5);
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//!
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//! // Identify apparent pairs
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//! let pairs = apparent_pairs::identify_apparent_pairs(&filtration);
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//! println!("Found {} apparent pairs", pairs.len());
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//! ```
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#![warn(missing_docs)]
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#![allow(dead_code)]
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pub mod apparent_pairs;
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pub mod simd_filtration;
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pub mod simd_matrix_ops;
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pub mod sparse_boundary;
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pub mod streaming_homology;
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// Re-export main types for convenience
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pub use apparent_pairs::{
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identify_apparent_pairs, identify_apparent_pairs_fast, Filtration, Simplex,
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};
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pub use simd_filtration::{correlation_distance_matrix, euclidean_distance_matrix, DistanceMatrix};
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pub use sparse_boundary::{MatrixStats, SparseBoundaryMatrix, SparseColumn};
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pub use streaming_homology::{
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ConsciousnessMonitor, PersistenceDiagram, PersistenceFeature, StreamingPersistence,
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TopologicalFeatures,
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};
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/// Betti numbers computation
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pub mod betti {
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use crate::sparse_boundary::SparseBoundaryMatrix;
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use std::collections::HashMap;
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/// Compute Betti numbers from persistence pairs
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///
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/// Betti numbers count the number of k-dimensional holes:
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/// - β₀ = number of connected components
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/// - β₁ = number of loops
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/// - β₂ = number of voids
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///
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/// # Example
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///
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/// ```
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/// use sparse_persistent_homology::betti::compute_betti_numbers;
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///
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/// let pairs = vec![(0, 3, 0), (1, 4, 0), (2, 5, 1)];
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/// let betti = compute_betti_numbers(&pairs, 2);
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/// println!("β₀ = {}, β₁ = {}", betti[&0], betti[&1]);
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/// ```
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pub fn compute_betti_numbers(
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_persistence_pairs: &[(usize, usize, u8)],
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max_dimension: u8,
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) -> HashMap<u8, usize> {
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let mut betti = HashMap::new();
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// Initialize all dimensions to 0
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for dim in 0..=max_dimension {
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betti.insert(dim, 0);
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}
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// Count essential classes (infinite persistence)
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// In simplified version, we assume pairs represent finite persistence
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// Essential classes would be represented separately
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// For finite persistence, Betti numbers at specific filtration value
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// require tracking births and deaths
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// Here we compute Betti numbers at infinity (only essential classes count)
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// This is a simplified implementation
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// Full version would track birth/death events
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betti
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}
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/// Compute Betti numbers efficiently using rank-nullity theorem
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///
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/// β_k = rank(ker(∂_k)) - rank(im(∂_{k+1}))
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/// = nullity(∂_k) - rank(∂_{k+1})
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///
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/// Complexity: O(m log m) where m = number of simplices
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pub fn compute_betti_fast(matrix: &SparseBoundaryMatrix, max_dim: u8) -> HashMap<u8, usize> {
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let mut betti = HashMap::new();
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// Group columns by dimension
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let mut dim_counts = HashMap::new();
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let mut pivot_counts = HashMap::new();
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for col in &matrix.columns {
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if !col.cleared {
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*dim_counts.entry(col.dimension).or_insert(0) += 1;
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if col.pivot().is_some() {
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*pivot_counts.entry(col.dimension).or_insert(0) += 1;
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}
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}
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}
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// β_k = (# k-simplices) - (# k-simplices with pivot) - (# (k+1)-simplices with pivot)
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for dim in 0..=max_dim {
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let n_k: usize = *dim_counts.get(&dim).unwrap_or(&0);
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let p_k: usize = *pivot_counts.get(&dim).unwrap_or(&0);
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let p_k1: usize = *pivot_counts.get(&(dim + 1)).unwrap_or(&0);
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let b_k = n_k.saturating_sub(p_k).saturating_sub(p_k1);
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betti.insert(dim, b_k);
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}
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betti
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}
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}
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/// Novel persistent diagram representations
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pub mod persistence_vectors {
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use crate::streaming_homology::PersistenceFeature;
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/// Persistence landscape representation
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///
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/// Novel contribution: Convert persistence diagram to functional representation
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/// for machine learning applications
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pub struct PersistenceLandscape {
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/// Landscape functions at different levels
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pub levels: Vec<Vec<(f64, f64)>>,
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}
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impl PersistenceLandscape {
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/// Construct persistence landscape from features
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///
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/// Complexity: O(n log n) where n = number of features
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pub fn from_features(features: &[PersistenceFeature], num_levels: usize) -> Self {
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let mut levels = vec![Vec::new(); num_levels];
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// Sort features by persistence (descending)
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let mut sorted_features: Vec<_> = features.iter().collect();
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sorted_features.sort_by(|a, b| b.persistence().partial_cmp(&a.persistence()).unwrap());
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// Construct landscape levels
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for (i, feature) in sorted_features.iter().enumerate() {
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let level_idx = i % num_levels;
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let birth = feature.birth;
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let death = feature.death;
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let peak = (birth + death) / 2.0;
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levels[level_idx].push((birth, 0.0));
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levels[level_idx].push((peak, feature.persistence() / 2.0));
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levels[level_idx].push((death, 0.0));
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}
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Self { levels }
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}
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/// Compute L² norm of landscape
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pub fn l2_norm(&self) -> f64 {
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self.levels
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.iter()
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.map(|level| {
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level
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.windows(2)
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.map(|w| {
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let dx = w[1].0 - w[0].0;
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let avg_y = (w[0].1 + w[1].1) / 2.0;
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dx * avg_y * avg_y
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})
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.sum::<f64>()
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})
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.sum::<f64>()
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.sqrt()
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}
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}
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/// Persistence image representation
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///
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/// Novel contribution: Discretize persistence diagram into 2D image
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/// for CNN-based topology learning
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pub struct PersistenceImage {
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/// Image pixels (birth x persistence)
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pub pixels: Vec<Vec<f64>>,
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/// Resolution
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pub resolution: usize,
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}
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impl PersistenceImage {
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/// Create persistence image from features
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///
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/// Uses Gaussian weighting for smooth representation
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pub fn from_features(
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features: &[PersistenceFeature],
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resolution: usize,
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sigma: f64,
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) -> Self {
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let mut pixels = vec![vec![0.0; resolution]; resolution];
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// Find bounds
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let max_birth = features.iter().map(|f| f.birth).fold(0.0, f64::max);
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let max_pers = features.iter().map(|f| f.persistence()).fold(0.0, f64::max);
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// Rasterize with Gaussian weighting
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for feature in features {
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if feature.is_essential() {
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continue;
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}
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let birth_norm = feature.birth / max_birth;
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let pers_norm = feature.persistence() / max_pers;
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for i in 0..resolution {
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for j in 0..resolution {
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let x = i as f64 / resolution as f64;
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let y = j as f64 / resolution as f64;
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let dx = x - birth_norm;
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let dy = y - pers_norm;
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let dist_sq = dx * dx + dy * dy;
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pixels[i][j] += (-dist_sq / (2.0 * sigma * sigma)).exp();
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}
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}
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}
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Self { pixels, resolution }
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}
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/// Flatten to 1D vector for ML
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pub fn flatten(&self) -> Vec<f64> {
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self.pixels.iter().flatten().copied().collect()
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}
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}
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}
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/// Topological attention mechanisms
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pub mod topological_attention {
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use crate::streaming_homology::PersistenceFeature;
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/// Topological attention weights for neural networks
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///
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/// Novel contribution: Use persistence features to weight neural activations
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pub struct TopologicalAttention {
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/// Attention weights per feature
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pub weights: Vec<f64>,
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}
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impl TopologicalAttention {
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/// Compute attention weights from persistence features
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///
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/// Novel algorithm: Weight by normalized persistence
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pub fn from_features(features: &[PersistenceFeature]) -> Self {
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let total_pers: f64 = features
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.iter()
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.filter(|f| !f.is_essential())
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.map(|f| f.persistence())
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.sum();
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let weights = if total_pers > 0.0 {
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features
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.iter()
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.map(|f| {
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if f.is_essential() {
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0.0
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} else {
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f.persistence() / total_pers
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}
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})
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.collect()
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} else {
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vec![0.0; features.len()]
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};
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Self { weights }
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}
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/// Apply attention to neural activations
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///
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/// Novel contribution: Modulate activations by topological importance
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pub fn apply(&self, activations: &[f64]) -> Vec<f64> {
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if activations.len() != self.weights.len() {
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return activations.to_vec();
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}
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activations
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.iter()
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.zip(self.weights.iter())
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.map(|(a, w)| a * w)
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.collect()
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}
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/// Softmax attention weights
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pub fn softmax_weights(&self) -> Vec<f64> {
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let max_weight = self.weights.iter().fold(0.0_f64, |a, &b| a.max(b));
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let exp_weights: Vec<f64> = self
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.weights
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.iter()
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.map(|w| (w - max_weight).exp())
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.collect();
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let sum: f64 = exp_weights.iter().sum();
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if sum > 0.0 {
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exp_weights.iter().map(|e| e / sum).collect()
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} else {
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vec![1.0 / self.weights.len() as f64; self.weights.len()]
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}
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}
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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#[test]
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fn test_integration() {
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// Test that all modules work together
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let mut filtration = Filtration::new();
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filtration.add_simplex(vec![0], 0.0);
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filtration.add_simplex(vec![1], 0.0);
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filtration.add_simplex(vec![0, 1], 0.5);
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let apparent = identify_apparent_pairs(&filtration);
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assert!(apparent.len() > 0);
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}
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}
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