# R/rsvd.R In Benli11/rSVD: Randomized Singular Value Decomposition

#### Documented in rsvd

#' @title  Randomized Singular Value Decomposition (rsvd).
#
#' @description The randomized SVD computes the near-optimal low-rank approximation of a rectangular matrix
#' using a fast probablistic algorithm.
#
#' @details
#' The singular value decomposition (SVD) plays an important role in data analysis, and scientific computing.
#' Given a rectangular \eqn{(m,n)} matrix \eqn{A}, and a target rank \eqn{k << min(m,n)},
#' the SVD factors the input matrix \eqn{A} as
#'
#' \deqn{ A  =  U_{k} diag(d_{k}) V_{k}^\top }{ A = U diag(d) t(V)}
#'
#' The \eqn{k} left singular vectors are the columns of the
#' real or complex unitary matrix \eqn{U}. The \eqn{k} right singular vectors are the columns
#' of the real or complex unitary matrix \eqn{V}. The \eqn{k} dominant singular values are the
#' entries of \eqn{d}, and non-negative and real numbers.
#'
#' \eqn{p} is an oversampling parameter to improve the approximation.
#' A value of at least 10 is recommended, and \eqn{p=10} is set by default.
#'
#' The parameter \eqn{q} specifies the number of power (subspace) iterations
#' to reduce the approximation error. The power scheme is recommended,
#' if the singular values decay slowly. In practice, 2 or 3 iterations
#' achieve good results, however, computing power iterations increases the
#' computational costs. The power scheme is set to \eqn{q=2} by default.
#'
#' If \eqn{k > (min(n,m)/4)}, a deterministic partial or truncated \code{\link{svd}}
#' algorithm might be faster.
#'
#'
#' @param A       array_like; \cr
#'                a real/complex \eqn{(m, n)} input matrix (or data frame) to be decomposed.
#'
#' @param k       integer; \cr
#'                specifies the target rank of the low-rank decomposition. \eqn{k} should satisfy \eqn{k << min(m,n)}.
#'
#' @param nu      integer, optional; \cr
#'                number of left singular vectors to be returned. \eqn{nu} must be between \eqn{0} and \eqn{k}.
#'
#' @param nv      integer, optional; \cr
#'                number of right singular vectors to be returned. \eqn{nv} must be between \eqn{0} and \eqn{k}.
#'
#' @param p       integer, optional; \cr
#'                oversampling parameter (by default \eqn{p=10}).
#'
#' @param q       integer, optional; \cr
#'                number of additional power iterations (by default \eqn{q=2}).
#'
#' @param sdist   string \eqn{c( 'unif', 'normal', 'rademacher')}, optional; \cr
#'                specifies the sampling distribution of the random test matrix: \cr
#'                		\eqn{'unif'} :  Uniform [-1,1]. \cr
#'                		\eqn{'normal}' (default) : Normal ~N(0,1). \cr
#'
#'@return \code{rsvd} returns a list containing the following three components:
#'\item{d}{  array_like; \cr
#'           singular values; vector of length \eqn{(k)}.
#'}
#'
#'\item{u}{  array_like; \cr
#'           left singular vectors; \eqn{(m, k)} or \eqn{(m, nu)} dimensional array.
#'}
#'
#'\item{v}{  array_like; \cr
#'           right singular vectors; \eqn{(n, k)} or \eqn{(n, nv)} dimensional array. \cr
#'}
#'
#' @note The singular vectors are not unique and only defined up to sign
#' (a constant of modulus one in the complex case). If a left singular vector
#' has its sign changed, changing the sign of the corresponding right vector
#' gives an equivalent decomposition.
#'
#'
#' @references
#' \itemize{
#'   \item  [1] N. Halko, P. Martinsson, and J. Tropp.
#'          "Finding structure with randomness: probabilistic
#'          algorithms for constructing approximate matrix
#'          decompositions" (2009).
#'          (available at arXiv \url{http://arxiv.org/abs/0909.4061}).
#'   \item  [2] N. B. Erichson, S. Voronin, S. Brunton, J. N. Kutz.
#'          "Randomized matrix decompositions using R" (2016).
#'          (available at arXiv \url{http://arxiv.org/abs/1608.02148}).
#' }
#'
#' @author N. Benjamin Erichson, \email{[email protected]}
#'
#'
#' @examples
#'library('rsvd')
#'
#'# Create a n x n Hilbert matrix of order n,
#'# with entries H[i,j] = 1 / (i + j + 1).
#'hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
#'H <- hilbert(n=50)
#'
#'# Low-rank (k=10) matrix approximation using rsvd
#'k=10
#'s <- rsvd(H, k=k)
#'Hre <- s$u %*% diag(s$d) %*% t(s$v) # matrix approximation #'print(100 * norm( H - Hre, 'F') / norm( H,'F')) # percentage error #'# Compare to truncated base svd #'s <- svd(H) #'Hre <- s$u[,1:k] %*% diag(s$d[1:k]) %*% t(s$v[,1:k]) # matrix approximation
#'print(100 * norm( H - Hre, 'F') / norm( H,'F')) # percentage error
#'

#' @export
rsvd <- function(A, k=NULL, nu=NULL, nv=NULL, p=10, q=2, sdist="normal") UseMethod("rsvd")

#' @export
rsvd.default <- function(A, k=NULL, nu=NULL, nv=NULL, p=10, q=2, sdist="normal") {
#*************************************************************************
#***        Author: N. Benjamin Erichson <[email protected]>        ***
#***                              <2015>                               ***
#***                       License: BSD 3 clause                       ***
#*************************************************************************

#Dim of input matrix
m <- nrow(A)
n <- ncol(A)

#Flipp matrix, if wide
if(m < n){
A <- H(A)
m <- nrow(A)
n <- ncol(A)
flipped <- TRUE
} else flipped <- FALSE

#Set target rank
if(is.null(k)) k = n
if(k > n) k <- n
if(is.character(k)) stop("Target rank is not valid!")
if(k < 1) stop("Target rank is not valid!")

#Set oversampling parameter
l <- round(k) + round(p)
if(l > n) l <- n
if(l < 1) stop("Target rank is not valid!")

#Check if array is real or complex
if(is.complex(A)) {
isreal <- FALSE
} else {
isreal <- TRUE
}

#Set number of singular vectors
if(is.null(nu)) nu <- k
if(is.null(nv)) nv <- k
if(nu < 0) nu <- 0
if(nv < 0) nv <- 0
if(nu > k) nu <- k
if(nv > k) nv <- k
if(flipped==TRUE) {
temp <- nu
nu <- nv
nv <- temp
}

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Generate a random sampling matrix O
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
O <- switch(sdist,
normal = matrix(stats::rnorm(l*n), n, l),
unif = matrix(stats::runif(l*n), n, l),
rademacher = matrix(sample(c(-1,1), (l*n), replace = TRUE, prob = c(0.5,0.5)), n, l),
stop("Selected sampling distribution is not supported!"))

if(isreal==FALSE) {
O <- O + switch(sdist,
normal = 1i * matrix(stats::rnorm(l*n), n, l),
unif = 1i * matrix(stats::runif(l*n), n, l),
rademacher = 1i * matrix(sample(c(-1,1), (l*n), replace = TRUE, prob = c(0.5,0.5)), n, l),
stop("Selected sampling distribution is not supported!"))
}

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Build sample matrix Y : Y = A * O
#Note: Y should approximate the range of A
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Y <- A %*% O
remove(O)

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Orthogonalize Y using economic QR decomposition: Y=QR
#If q > 0 perfrom q subspace iterations
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if( q > 0 ) {
for( i in 1:q) {
Y <- qr.Q( qr(Y, complete = FALSE) , complete = FALSE )
Z <- crossprod_help(A , Y )
Z <- qr.Q( qr(Z, complete = FALSE) , complete = FALSE )
Y <- A %*% Z
}#End for
remove(Z)
}#End if

Q <- qr.Q( qr(Y, complete = FALSE) , complete = FALSE )
remove(Y)

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Project the data matrix a into a lower dimensional subspace
#B := Q.T * A
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
B <- crossprod_help(Q , A )

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Singular Value Decomposition
#Note: B =: U * S * Vt
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
rsvdObj <- svd(B, nu=nu, nv=nv) # Compute SVD
rsvdObj$d <- rsvdObj$d[1:k] # Truncate singular values

if(nu != 0) rsvdObj$u <- Q %*% rsvdObj$u # Recover left singular vectors

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Flipp SVD back
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if(flipped == TRUE) {
u_temp <- rsvdObj$u rsvdObj$u <- rsvdObj$v rsvdObj$v <- u_temp
}

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Return
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if(nu == 0){ rsvdObj$u <- NULL} if(nv == 0){ rsvdObj$v <- NULL}
class(rsvdObj) <- "rsvd"
return(rsvdObj)

} # End rsvd
`
Benli11/rSVD documentation built on Nov. 6, 2018, 10:46 p.m.