irlba: svdr – R documentation

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svdr

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

Description

The randomized method for truncated SVD by P. G. Martinsson and colleagues finds a few approximate largest singular values and corresponding singular vectors of a sparse or dense matrix. It is a fast and memory-efficient way to compute a partial SVD, similar in performance for many problems to irlba. The svdr method is a block method and may produce more accurate estimations with less work for problems with clustered large singular values (see the examples). In other problems, irlba may exhibit faster convergence.

Usage

svdr(x, k, tol = 1e-05, it = 100L, extra = min(10L, dim(x) - k),
  center = NULL, Q = NULL, return.Q = FALSE)

Arguments

`x`	numeric real- or complex-valued matrix or real-valued sparse matrix.
`k`	dimension of subspace to estimate (number of approximate singular values to compute).
`tol`	stop iteration when the largest absolute relative change in estimated singular values from one iteration to the next falls below this value.
`it`	maximum number of algorithm iterations.
`extra`	number of extra vectors of dimension `ncol(x)`, larger values generally improve accuracy (with increased computational cost).
`center`	optional column centering vector whose values are implicitly subtracted from each column of `A` without explicitly forming the centered matrix (preserving sparsity). Optionally specify `center=TRUE` as shorthand for `center=colMeans(x)`. Use for efficient principal components computation.
`Q`	optional initial random matrix, defaults to a matrix of size `ncol(x)` by `k + extra` with entries sampled from a normal random distribution.
`return.Q`	if `TRUE` return the `Q` matrix for restarting (see examples).

Details

Also see an alternate implementation (rsvd) of this method by N. Benjamin Erichson in the https://cran.r-project.org/package=rsvd package.

Value

Returns a list with entries:

d:: k approximate singular values
u:: k approximate left singular vectors
v:: k approximate right singular vectors
mprod:: total number of matrix products carried out
Q:: optional subspace matrix (when return.Q=TRUE)

References

Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions N. Halko, P. G. Martinsson, J. Tropp. Sep. 2009.

Examples

set.seed(1)

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

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

# Compare with irlba
irlba(A, 3)$d

## Not run: 
# A problem with clustered large singular values where svdr out-performs irlba.
tprolate <- function(n, w=0.25)
{
  a <- rep(0, n)
  a[1] <- 2 * w
  a[2:n] <- sin( 2 * pi * w * (1:(n-1)) ) / ( pi * (1:(n-1)) )
  toeplitz(a)
}

x <- tprolate(512)
set.seed(1)
tL <- system.time(L <- irlba(x, 20))
tR <- system.time(R <- svdr(x, 20))
S <- svd(x)
plot(S$d)
data.frame(time=c(tL[3], tR[3]),
           error=sqrt(c(crossprod(L$d - S$d[1:20]), crossprod(R$d - S$d[1:20]))),
           row.names=c("IRLBA", "Randomized SVD"))

# But, here is a similar problem with clustered singular values where svdr
# doesn't out-perform irlba as easily...clusters of singular values are,
# in general, very hard to deal with!
# (This example based on https://github.com/bwlewis/irlba/issues/16.)
set.seed(1)
s <- svd(matrix(rnorm(200 * 200), 200))
x <- s$u %*% (c(exp(-(1:100)^0.3) * 1e-12 + 1, rep(0.5, 100)) * t(s$v))
tL <- system.time(L <- irlba(x, 5))
tR <- system.time(R <- svdr(x, 5))
S <- svd(x)
plot(S$d)
data.frame(time=c(tL[3], tR[3]),
           error=sqrt(c(crossprod(L$d - S$d[1:5]), crossprod(R$d - S$d[1:5]))),
           row.names=c("IRLBA", "Randomized SVD"))

## End(Not run)

irlba

Fast Truncated Singular Value Decomposition and Principal Components Analysis for Large Dense and Sparse Matrices

v2.3.3

GPL-3

Authors

Jim Baglama [aut, cph], Lothar Reichel [aut, cph], B. W. Lewis [aut, cre, cph]

Initial release

2019-02-04