Implement Advanced Dimensionality Reduction Algorithms

[ianfd]

Description

Extend single-algebra with t-SNE and UMAP dimensionality reduction algorithms to complement the existing PCA implementation.

Objectives

• Add mathematical foundations for t-SNE and UMAP
• Optimize implementations for performance with large matrices
• Ensure algorithms work efficiently with sparse data structures

Key Components to Implement

t-SNE Implementation

• Implement core t-SNE algorithm using nalgebra
• Add Barnes-Hut approximation for scaling to large datasets
• Support various distance metrics (Euclidean, cosine, etc.)
• Optimize gradient calculations for performance

UMAP Implementation

• Evaluate integration with fast-umap (https://github.com/eugenehp/fast-umap)
• If needed, implement core UMAP algorithm based on reference
• Add fuzzy topological representation functionality
• Implement efficient nearest neighbor computations

Shared Components

• Distance matrix calculation optimizations
• Stochastic gradient descent improvements
• Neighbor graph construction utilities
• Progress reporting for long-running calculations

Integration Points

• Should integrate with existing PCA implementation
• Must work with both dense and sparse matrix types
• Support for both f32 and f64 precision

Technical Notes

• Consider SIMD optimization for distance calculations
• Implement multithreading via Rayon for performance-critical sections
• Ensure memory efficiency for large datasets