# Find a few approximate largest singular values and corresponding singular vectors of a matrix. 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. ```r 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). ## Returns 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`) ## Details Also see an alternate implementation (`rsvd`) of this method by N. Benjamin Erichson in the https://cran.r-project.org/package=rsvd package. ## Examples ```r 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) ``` ## References Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions N. Halko, P. G. Martinsson, J. Tropp. Sep. 2009. ## See Also `irlba`, `svd`, `rsvd` in the rsvd package

svdr function