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Linear Algebra Operations

single-algebra provides robust linear algebra capabilities built on multiple backend options. These operations form the foundation for dimensionality reduction, matrix decomposition, and other advanced analytical techniques essential for working with high-dimensional data.

SVD (Singular Value Decomposition)​

SVD is a fundamental matrix factorization technique that decomposes a matrix into three matrices: U, Σ, and V^T. single-algebra implements SVD with multiple backend options:

  • LAPACK-based SVD: Leverages the industry-standard LAPACK library through the nalgebra-lapack crate with OpenBLAS backend for high performance.
  • Faer-based SVD: Uses the Faer library, a pure Rust implementation optimized for modern CPU architectures.
  • Single-SVDLib: A specialized implementation for large sparse matrices.

All SVD implementations expose a consistent interface for retrieving:

  • Left singular vectors (U)
  • Singular values (Σ)
  • Right singular vectors (V^T)
  • Matrix reconstruction

PCA (Principal Component Analysis)​

PCA is implemented as a higher-level abstraction built on SVD, with additional features:

  • Standard PCA: Full-matrix PCA implementation with optional centering and scaling
  • Sparse PCA: Specialized version optimized for sparse matrices
  • Configurable Components: Control the number of principal components to extract
  • Variance Analysis: Calculate explained variance ratios and cumulative explained variance

The modular architecture allows for extension with different SVD implementations via the SVDImplementation trait.

Matrix Transformations​

  • Matrix Creation and Conversion: Utilities to create matrices from various data sources
  • Matrix-Matrix Multiplication: Optimized for both sparse and dense representations
  • Matrix-Vector Operations: Efficient implementations of common operations

Integration with External Libraries​

single-algebra provides seamless integration with multiple linear algebra ecosystems:

  • nalgebra: Core integration for general-purpose linear algebra operations
  • ndarray: Integration for n-dimensional array processing
  • Faer: Modern, SIMD-optimized implementations
  • BLAS/LAPACK: Industry-standard high-performance routines through OpenBLAS

Builder Pattern for Configurability​

Components like PCA use a builder pattern for flexible configuration:

// Example of the PCA builder pattern (not executable code)
let pca = PCABuilder::new(svd_implementation)
    .n_components(10)
    .center(true)
    .scale(false)
    .build();

This approach allows for clear, flexible configuration while providing sensible defaults.

Performance Considerations​

  • Backend Selection: Different backends offer trade-offs between pure-Rust implementation (faer) and optimized C implementations (LAPACK)
  • Sparse Matrix Support: Optimized approaches for sparse data that avoid materializing dense representations
  • Memory Efficiency: Implementations that minimize temporary allocations
  • Parallelization: Parallel implementations of key operations for multi-core processors

Application Areas​

These linear algebra operations serve as building blocks for:

  • Dimensionality reduction in high-dimensional data
  • Feature extraction in machine learning pipelines
  • Signal processing and data compression
  • Network and graph analysis
  • Statistical modeling and inference

The modular design of single-algebra allows users to select the most appropriate implementation for their specific use case, balancing accuracy, performance, and memory requirements.