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

svds {RSpectra}

R Documentation

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 base package.
`dgeMatrix`	General matrix, equivalent to `matrix`, defined in Matrix package.
`dgCMatrix`	Column oriented sparse matrix, defined in Matrix package.
`dgRMatrix`	Row oriented sparse matrix, defined in Matrix package.
`dsyMatrix`	Symmetrix matrix, defined in Matrix package.
`function`	Implicitly specify the matrix through two functions that calculate f(x) = A x* and g(x) = A' x. See section Function Interface* for details.

Note that when A is symmetric, SVD reduces to eigen decomposition, so you may consider using eigs() instead.

Usage

svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'matrix'
svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'dgeMatrix'
svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'dgCMatrix'
svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'dgRMatrix'
svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'dsyMatrix'
svds(A, k, nu = k, nv = k, opts = list(), ...)

## S3 method for class 'function'
svds(A, k, nu = k, nv = k, opts = list(), ..., Atrans,
  dim, args = NULL)

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.
`Atrans`	Only used when `A` is a function. `A` is a function that calculates the matrix multiplication A x, 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.
`...`	Arguments for specialized S3 function calls, for example `Atrans`, `dim` and `args`.

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 <= 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.

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.

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 A * x 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.

Author(s)

Yixuan Qiu <http://statr.me>

Examples

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)

[Package RSpectra version 0.12-0 Index]