svdlibrs —

从SVDLIBC移植LAS2到Rust

一个计算稀疏矩阵（通常是大型稀疏矩阵）的SVD的库

这是对Doug Rohde的SVDLIBC中实现的算法的功能性移植（主要是翻译）

此库使用Lanczos算法对稀疏输入矩阵进行奇异值分解，并返回分解为ndarray组件。

用法

输入：稀疏矩阵（CSR、CSC或COO）

输出：分解 U、S、V，其中U、V是Array2，而S是Array1，包装在Result<SvdRec，SvdLibError>

快速入门

有3个方便的方法来处理常见用例

1. svd — 简单地计算SVD

2. svd_dim — 计算SVD并提供所需的维度数

3. svd_dim_seed — 计算SVD并提供所需的维度数和一个固定的种子到LAS2算法（算法使用随机向量初始化，如果没有提供，将生成一个内部种子）

use svdlibrs::svd;

// SVD on a Compressed Sparse Row matrix
let svd = svd(&csr)?;

use svdlibrs::svd_dim;

// SVD on a Compressed Sparse Column matrix specifying the desired dimensions, 3 in this example
let svd = svd_dim(&csc, 3)?;

use svdlibrs::svd_dim_seed;

// SVD on a Coordinate-form matrix requesting the
// dimensions and supplying a fixed seed to the LAS2 algorithm
let svd = svd_dim_seed(&coo, dimensions, 12345)?;

SVD和信息诊断返回在SvdRec

pub struct SvdRec {
    pub d: usize,        // Dimensionality (rank), the number of rows of both ut, vt and the length of s
    pub ut: Array2<f64>, // Transpose of left singular vectors, the vectors are the rows of ut
    pub s: Array1<f64>,  // Singular values (length d)
    pub vt: Array2<f64>, // Transpose of right singular vectors, the vectors are the rows of vt
    pub diagnostics: Diagnostics, // Computational diagnostics
}

pub struct Diagnostics {
    pub non_zero: usize,   // Number of non-zeros in the input matrix
    pub dimensions: usize, // Number of dimensions attempted (bounded by matrix shape)
    pub iterations: usize, // Number of iterations attempted (bounded by dimensions and matrix shape)
    pub transposed: bool,  // True if the matrix was transposed internally
    pub lanczos_steps: usize,          // Number of Lanczos steps performed
    pub ritz_values_stabilized: usize, // Number of ritz values
    pub significant_values: usize,     // Number of significant values discovered
    pub singular_values: usize,        // Number of singular values returned
    pub end_interval: [f64; 2], // Left, Right end of interval containing unwanted eigenvalues
    pub kappa: f64,             // Relative accuracy of ritz values acceptable as eigenvalues
    pub random_seed: u32,       // Random seed provided or the seed generated
}

svdLAS2方法提供了以下参数控制

use svdlibrs::{svd, svd_dim, svd_dim_seed, svdLAS2, SvdRec};

let svd: SvdRec = svdLAS2(
    &matrix,      // sparse matrix (nalgebra_sparse::{csr,csc,coo}
    dimensions,   // upper limit of desired number of dimensions
                  // supplying 0 will use the input matrix shape to determine dimensions
    iterations,   // number of algorithm iterations
                  // supplying 0 will use the input matrix shape to determine iterations
    end_interval, // left, right end of interval containing unwanted eigenvalues,
                  // typically small values centered around zero
                  /// set to [-1.0e-30, 1.0e-30] for convenience methods svd(), svd_dim(), svd_dim_seed()
    kappa,        // relative accuracy of ritz values acceptable as eigenvalues
                  /// set to 1.0e-6 for convenience methods svd(), svd_dim(), svd_dim_seed()
    random_seed,  // a supplied seed if > 0, otherwise an internal seed will be generated
)?;

SVD示例

使用R进行SVD

$ Rscript -e 'options(digits=12);m<-matrix(1:9,nrow=3)^2;print(m);r<-svd(m);print(r);r$u%*%diag(r$d)%*%t(r$v)'

• The input matrix: M
     [,1] [,2] [,3]
[1,]    1   16   49
[2,]    4   25   64
[3,]    9   36   81

• The diagonal matrix (singular values): S
$d
[1] 123.676578742544   6.084527896514   0.287038004183

• The left singular vectors: U
$u
                [,1]            [,2]            [,3]
[1,] -0.415206840886 -0.753443585619 -0.509829424976
[2,] -0.556377565194 -0.233080213641  0.797569820742
[3,] -0.719755016815  0.614814099788 -0.322422608499

• The right singular vectors: V
$v
                 [,1]            [,2]            [,3]
[1,] -0.0737286909592  0.632351847728 -0.771164846712
[2,] -0.3756889918995  0.698691000150  0.608842071210
[3,] -0.9238083467338 -0.334607272761 -0.186054055373

• Recreating the original input matrix: r$u %*% diag(r$d) %*% t(r$v)
     [,1] [,2] [,3]
[1,]    1   16   49
[2,]    4   25   64
[3,]    9   36   81

使用svdlibrs进行SVD

• Cargo.toml依赖项

[dependencies]
svdlibrs = "0.5.1"
nalgebra-sparse = "0.9.0"
ndarray = "0.15.6"

extern crate ndarray;
use ndarray::prelude::*;
use nalgebra_sparse::{coo::CooMatrix, csc::CscMatrix};
use svdlibrs::svd_dim_seed;

fn main() {
    // create a CscMatrix from a CooMatrix
    // use the same matrix values as the R example above
    //      [,1] [,2] [,3]
    // [1,]    1   16   49
    // [2,]    4   25   64
    // [3,]    9   36   81
    let mut coo = CooMatrix::<f64>::new(3, 3);
    coo.push(0, 0, 1.0); coo.push(0, 1, 16.0); coo.push(0, 2, 49.0);
    coo.push(1, 0, 4.0); coo.push(1, 1, 25.0); coo.push(1, 2, 64.0);
    coo.push(2, 0, 9.0); coo.push(2, 1, 36.0); coo.push(2, 2, 81.0);

    // our input
    let csc = CscMatrix::from(&coo);

    // compute the svd
    // 1. supply 0 as the dimension (requesting max)
    // 2. supply a fixed seed so outputs are repeatable between runs
    let svd = svd_dim_seed(&csc, 0, 3141).unwrap();

    // svd.d dimensions were found by the algorithm
    // svd.ut is a 2-d array holding the left vectors
    // svd.vt is a 2-d array holding the right vectors
    // svd.s is a 1-d array holding the singular values
    // assert the shape of all results in terms of svd.d
    assert_eq!(svd.d, 3);
    assert_eq!(svd.d, svd.ut.nrows());
    assert_eq!(svd.d, svd.s.dim());
    assert_eq!(svd.d, svd.vt.nrows());

    // show transposed output
    println!("svd.d = {}\n", svd.d);
    println!("U =\n{:#?}\n", svd.ut.t());
    println!("S =\n{:#?}\n", svd.s);
    println!("V =\n{:#?}\n", svd.vt.t());

    // Note: svd.ut & svd.vt are returned in transposed form
    // M = USV*
    let m_approx = svd.ut.t().dot(&Array2::from_diag(&svd.s)).dot(&svd.vt);
    assert_eq!(svd.recompose(), m_approx);

    // assert computed values are an acceptable approximation
    let epsilon = 1.0e-12;
    assert!((m_approx[[0, 0]] - 1.0).abs() < epsilon);
    assert!((m_approx[[0, 1]] - 16.0).abs() < epsilon);
    assert!((m_approx[[0, 2]] - 49.0).abs() < epsilon);
    assert!((m_approx[[1, 0]] - 4.0).abs() < epsilon);
    assert!((m_approx[[1, 1]] - 25.0).abs() < epsilon);
    assert!((m_approx[[1, 2]] - 64.0).abs() < epsilon);
    assert!((m_approx[[2, 0]] - 9.0).abs() < epsilon);
    assert!((m_approx[[2, 1]] - 36.0).abs() < epsilon);
    assert!((m_approx[[2, 2]] - 81.0).abs() < epsilon);

    assert!((svd.s[0] - 123.676578742544).abs() < epsilon);
    assert!((svd.s[1] - 6.084527896514).abs() < epsilon);
    assert!((svd.s[2] - 0.287038004183).abs() < epsilon);
}

输出

svd.d = 3

U =
[[-0.4152068408862081, -0.7534435856189199, -0.5098294249756481],
 [-0.556377565193878, -0.23308021364108839, 0.7975698207417085],
 [-0.719755016814907, 0.6148140997884891, -0.3224226084985998]], shape=[3, 3], strides=[1, 3], layout=Ff (0xa), const ndim=2

S =
[123.67657874254405, 6.084527896513759, 0.2870380041828973], shape=[3], strides=[1], layout=CFcf (0xf), const ndim=1

V =
[[-0.07372869095916511, 0.6323518477280158, -0.7711648467120451],
 [-0.3756889918994792, 0.6986910001499903, 0.6088420712097343],
 [-0.9238083467337805, -0.33460727276072516, -0.18605405537270261]], shape=[3, 3], strides=[1, 3], layout=Ff (0xa), const ndim=2

以上示例的完整结果看起来像这样

svd = Ok(
    SvdRec {
       d: 3,
       ut: [[-0.4152068408862081, -0.556377565193878, -0.719755016814907],
            [-0.7534435856189199, -0.23308021364108839, 0.6148140997884891],
            [-0.5098294249756481, 0.7975698207417085, -0.3224226084985998]], shape=[3, 3], strides=[3, 1], layout=Cc (0x5), const ndim=2,
       s: [123.67657874254405, 6.084527896513759, 0.2870380041828973], shape=[3], strides=[1], layout=CFcf (0xf), const ndim=1,
       vt: [[-0.07372869095916511, -0.3756889918994792, -0.9238083467337805],
            [0.6323518477280158, 0.6986910001499903, -0.33460727276072516],
            [-0.7711648467120451, 0.6088420712097343, -0.18605405537270261]], shape=[3, 3], strides=[3, 1], layout=Cc (0x5), const ndim=2,
        diagnostics: Diagnostics {
            non_zero: 9,
            dimensions: 3,
            iterations: 3,
            transposed: false,
            lanczos_steps: 3,
            ritz_values_stabilized: 3,
            significant_values: 3,
            singular_values: 3,
            end_interval: [
                -1e-30,
                1e-30,
            ],
            kappa: 1e-6,
            random_seed: 3141,
        },
    },
)