irlba: Find a few approximate singular values and corresponding singular vectors of a matrix.

Description

The augmented implicitly restarted Lanczos bidiagonalization algorithm (IRLBA) finds a few approximate largest (or, optionally, smallest) singular values and corresponding singular vectors of a sparse or dense matrix using a method of Baglama and Reichel. It is a fast and memory-efficient way to compute a partial SVD.

Usage

irlba(A, nv = 5, nu = nv, maxit = 1000, work = nv + 7, reorth = TRUE,
  tol = 1e-05, v = NULL, right_only = FALSE, verbose = FALSE,
  scale = NULL, center = NULL, shift = NULL, mult = NULL,
  fastpath = TRUE, svtol = tol, smallest = FALSE, ...)

Value

Returns a list with entries:

d:: max(nu, nv) approximate singular values
u:: nu approximate left singular vectors (only when right_only=FALSE)
v:: nv approximate right singular vectors
iter:: The number of Lanczos iterations carried out
mprod:: The total number of matrix vector products carried out

Arguments

A: numeric real- or complex-valued matrix or real-valued sparse matrix.
nv: number of right singular vectors to estimate.
nu: number of left singular vectors to estimate (defaults to nv).
maxit: maximum number of iterations.
work: working subspace dimension, larger values can speed convergence at the cost of more memory use.
reorth: if TRUE, apply full reorthogonalization to both SVD bases, otherwise only apply reorthogonalization to the right SVD basis vectors; the latter case is cheaper per iteration but, overall, may require more iterations for convergence. Automatically TRUE when fastpath=TRUE (see below).
tol: convergence is determined when $‖ A^{T} U - V S ‖ < t o l ‖ A ‖$ , and when the maximum relative change in estimated singular values from one iteration to the next is less than svtol = tol (see svtol below), where the spectral norm ||A|| is approximated by the largest estimated singular value, and U, V, S are the matrices corresponding to the estimated left and right singular vectors, and diagonal matrix of estimated singular values, respectively.
v: optional starting vector or output from a previous run of irlba used to restart the algorithm from where it left off (see the notes).
right_only: logical value indicating return only the right singular vectors (TRUE) or both sets of vectors (FALSE). The right_only option can be cheaper to compute and use much less memory when nrow(A) >> ncol(A) but note that obtained solutions typically lose accuracy due to lack of re-orthogonalization in the algorithm and that right_only = TRUE sets fastpath = FALSE (only use this option for really large problems that run out of memory and when nrow(A) >> ncol(A)). Consider increasing the work option to improve accuracy with right_only=TRUE.
verbose: logical value that when TRUE prints status messages during the computation.
scale: optional column scaling vector whose values divide each column of A; must be as long as the number of columns of A (see notes).
center: optional column centering vector whose values are subtracted from each column of A; must be as long as the number of columns of A and may not be used together with the deflation options below (see notes).
shift: optional shift value (square matrices only, see notes).
mult: DEPRECATED optional custom matrix multiplication function (default is %*%, see notes).
fastpath: try a fast C algorithm implementation if possible; set fastpath=FALSE to use the reference R implementation. See the notes for more details.
svtol: additional stopping tolerance on maximum allowed absolute relative change across each estimated singular value between iterations. The default value of this parameter is to set it to tol. You can set svtol=Inf to effectively disable this stopping criterion. Setting svtol=Inf allows the method to terminate on the first Lanczos iteration if it finds an invariant subspace, but with less certainty that the converged subspace is the desired one. (It may, for instance, miss some of the largest singular values in difficult problems.)
smallest: set smallest=TRUE to estimate the smallest singular values and associated singular vectors. WARNING: this option is somewhat experimental, and may produce poor estimates for ill-conditioned matrices.
...: optional additional arguments used to support experimental and deprecated features.

References

Baglama, James, and Lothar Reichel. "Augmented implicitly restarted Lanczos bidiagonalization methods." SIAM Journal on Scientific Computing 27.1 (2005): 19-42.

Examples

Run this code

set.seed(1)

A <- matrix(runif(400), nrow=20)
S <- irlba(A, 3)
S$d

# Compare with svd
svd(A)$d[1:3]

# Restart the algorithm to compute more singular values
# (starting with an existing solution S)
S1 <- irlba(A, 5, v=S)

# Estimate smallest singular values
irlba(A, 3, smallest=TRUE)$d

#Compare with
tail(svd(A)$d, 3)

# Principal components (see also prcomp_irlba)
P <- irlba(A, nv=1, center=colMeans(A))

# Compare with prcomp and prcomp_irlba (might vary up to sign)
cbind(P$v,
      prcomp(A)$rotation[, 1],
      prcomp_irlba(A)$rotation[, 1])

# A custom matrix multiplication function that scales the columns of A
# (cf the scale option). This function scales the columns of A to unit norm.
col_scale <- sqrt(apply(A, 2, crossprod))
setClass("scaled_matrix", contains="matrix", slots=c(scale="numeric"))
setMethod("%*%", signature(x="scaled_matrix", y="numeric"),
   function(x ,y) x@.Data %*% (y / x@scale))
setMethod("%*%", signature(x="numeric", y="scaled_matrix"),
   function(x ,y) (x %*% y@.Data) / y@scale)
a <- new("scaled_matrix", A, scale=col_scale)
irlba(a, 3)$d

# Compare with:
svd(sweep(A, 2, col_scale, FUN=`/`))$d[1:3]

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