svds: Find the Largest k Singular Values/Vectors of a Matrix

Description

Given an \(m\) by \(n\) matrix \(A\), function svds() can find its largest \(k\) singular values and the corresponding singular vectors. It is also called the Truncated SVD or Partial SVD since it only calculates a subset of the whole singular triplets.

Currently svds() supports matrices of the following classes:

`matrix`	The most commonly used matrix type, defined in the base package.
`dgeMatrix`	General matrix, equivalent to `matrix`, defined in the Matrix package.
`dgCMatrix`	Column oriented sparse matrix, defined in the Matrix package.
`dgRMatrix`	Row oriented sparse matrix, defined in the Matrix package.
`dsyMatrix`	Symmetrix matrix, defined in the Matrix package.
`dsCMatrix`	Symmetric column oriented sparse matrix, defined in the Matrix package.
`dsRMatrix`	Symmetric row oriented sparse matrix, defined in the Matrix package.
`function`	Implicitly specify the matrix through two functions that calculate \(f(x)=Ax\) and \(g(x)=A'x\). See section Function Interface for details.

Note that when \(A\) is symmetric and positive semi-definite, SVD reduces to eigen decomposition, so you may consider using eigs() instead. When \(A\) is symmetric but not necessarily positive semi-definite, the left and right singular vectors are the same as the left and right eigenvectors, but the singular values and eigenvalues will not be the same. In particular, if \(\lambda\) is a negative eigenvalue of \(A\), then \(|\lambda|\) will be the corresponding singular value.

Usage

svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for matrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dgeMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dgCMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dgRMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dsyMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dsCMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for dsRMatrix
svds(A, k, nu = k, nv = k, opts = list(), ...)
# S3 method for `function`
svds(A, k, nu = k, nv = k, opts = list(), ..., Atrans, dim, args = NULL)

Value

A list with the following components:

d: A vector of the computed singular values.
u: An m by nu matrix whose columns contain the left singular vectors. If nu == 0, NULL will be returned.
v: An n by nv matrix whose columns contain the right singular vectors. If nv == 0, NULL will be returned.
nconv: Number of converged singular values.
niter: Number of iterations used.
nops: Number of matrix-vector multiplications used.

Arguments

A: The matrix whose truncated SVD is to be computed.
k: Number of singular values requested.
nu: Number of left singular vectors to be computed. This must be between 0 and k.
nv: Number of right singular vectors to be computed. This must be between 0 and k.
opts: Control parameters related to the computing algorithm. See Details below.
...: Arguments for specialized S3 function calls, for example Atrans, dim and args.
Atrans: Only used when A is a function. A is a function that calculates the matrix multiplication \(Ax\), and Atrans is a function that calculates the transpose multiplication \(A'x\).
dim: Only used when A is a function, to specify the dimension of the implicit matrix. A vector of length two.
args: Only used when A is a function. This argument will be passed to the A and Atrans functions.

Function Interface

The matrix \(A\) can be specified through two functions with the following definitions

A <- function(x, args)
{
    ## should return A %*% x
}
Atrans <- function(x, args)
{
    ## should return t(A) %*% x
}

They receive a vector x as an argument and returns a vector of the proper dimension. These two functions should have the effect of calculating \(Ax\) and \(A'x\) respectively, and extra arguments can be passed in through the args parameter. In svds(), user should also provide the dimension of the implicit matrix through the argument dim.

The function interface does not support the center and scale parameters in opts.

Author

Yixuan Qiu <https://statr.me>

Details

The opts argument is a list that can supply any of the following parameters:

ncv: Number of Lanzcos basis vectors to use. More vectors will result in faster convergence, but with greater memory use. ncv must be satisfy \(k < ncv \le p\) where p = min(m, n). Default is min(p, max(2*k+1, 20)).
tol: Precision parameter. Default is 1e-10.
maxitr: Maximum number of iterations. Default is 1000.
center: Either a logical value (TRUE/FALSE), or a numeric vector of length \(n\). If a vector \(c\) is supplied, then SVD is computed on the matrix \(A - 1c'\), in an implicit way without actually forming this matrix. center = TRUE has the same effect as center = colMeans(A). Default is FALSE.
scale: Either a logical value (TRUE/FALSE), or a numeric vector of length \(n\). If a vector \(s\) is supplied, then SVD is computed on the matrix \((A - 1c')S\), where \(c\) is the centering vector and \(S = diag(1/s)\). If scale = TRUE, then the vector \(s\) is computed as the column norm of \(A - 1c'\). Default is FALSE.

Examples

Run this code

m = 100
n = 20
k = 5
set.seed(111)
A = matrix(rnorm(m * n), m)

svds(A, k)
svds(t(A), k, nu = 0, nv = 3)

## Sparse matrices
library(Matrix)
A[sample(m * n, m * n / 2)] = 0
Asp1 = as(A, "dgCMatrix")
Asp2 = as(A, "dgRMatrix")

svds(Asp1, k)
svds(Asp2, k, nu = 0, nv = 0)

## Function interface
Af = function(x, args)
{
    as.numeric(args %*% x)
}

Atf = function(x, args)
{
    as.numeric(crossprod(args, x))
}

svds(Af, k, Atrans = Atf, dim = c(m, n), args = Asp1)

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