Nothing
#' @title Randomized robust principal component analysis (rrpca).
#
#' @description Robust principal components analysis separates a matrix into a low-rank plus sparse component.
#
#' @details
#' Robust principal component analysis (RPCA) is a method for the robust seperation of a
#' a rectangular \eqn{(m,n)} matrix \eqn{A} into a low-rank component \eqn{L} and a
#' sparse comonent \eqn{S}:
#'
#' \deqn{A = L + S}
#'
#' To decompose the matrix, we use the inexact augmented Lagrange multiplier
#' method (IALM). The algorithm can be used in combination with either the randomized or deterministic SVD.
#'
#'
#' @param A array_like; \cr
#' a real \eqn{(m, n)} input matrix (or data frame) to be decomposed. \cr
#' na.omit is applied, if the data contain \eqn{NA}s.
#'
#' @param lambda scalar, optional; \cr
#' tuning parameter (default \eqn{lambda = max(m,n)^-0.5}).
#'
#' @param maxiter integer, optional; \cr
#' maximum number of iterations (default \eqn{maxiter = 50}).
#'
#' @param tol scalar, optional; \cr
#' precision parameter (default \eqn{tol = 1.0e-5}).
#'
#' @param p integer, optional; \cr
#' oversampling parameter for \eqn{rsvd} (default \eqn{p=10}), see \code{\link{rsvd}}.
#'
#' @param q integer, optional; \cr
#' number of additional power iterations for \eqn{rsvd} (default \eqn{q=2}), see \code{\link{rsvd}}.
#'
#' @param trace bool, optional; \cr
#' print progress.
#'
#' @param rand bool, optional; \cr
#' if (\eqn{TRUE}), the \eqn{rsvd} routine is used, otherwise \eqn{svd} is used.
#'
#'
#' @return \code{rrpca} returns a list containing the following components:
#' \describe{
#' \item{L}{ array_like; \cr
#' low-rank component; \eqn{(m, n)} dimensional array.
#' }
#' \item{S}{ array_like \cr
#' sparse component; \eqn{(m, n)} dimensional array.
#' }
#'}
#'
#' @author N. Benjamin Erichson, \email{erichson@berkeley.edu}
#'
#'
#' @references
#' \itemize{
#' \item [1] N. B. Erichson, S. Voronin, S. L. Brunton and J. N. Kutz. 2019.
#' Randomized Matrix Decompositions Using {R}.
#' Journal of Statistical Software, 89(11), 1-48.
#' \doi{10.18637/jss.v089.i11}.
#'
#' \item [2] Lin, Zhouchen, Minming Chen, and Yi Ma.
#' "The augmented lagrange multiplier method for exact
#' recovery of corrupted low-rank matrices." (2010).
#' (available at arXiv \url{https://arxiv.org/abs/1009.5055}).
#' }
#'
#' @examples
#' library('rsvd')
#'
#' # Create toy video
#' # background frame
#' xy <- seq(-50, 50, length.out=100)
#' mgrid <- list( x=outer(xy*0,xy,FUN="+"), y=outer(xy,xy*0,FUN="+") )
#' bg <- 0.1*exp(sin(-mgrid$x**2-mgrid$y**2))
#' toyVideo <- matrix(rep(c(bg), 100), 100*100, 100)
#'
#' # add moving object
#' for(i in 1:90) {
#' mobject <- matrix(0, 100, 100)
#' mobject[i:(10+i), 45:55] <- 0.2
#' toyVideo[,i] = toyVideo[,i] + c( mobject )
#' }
#'
#' # Foreground/Background separation
#' out <- rrpca(toyVideo, trace=TRUE)
#'
#' # Display results of the seperation for the 10th frame
#' par(mfrow=c(1,4))
#' image(matrix(bg, ncol=100, nrow=100)) #true background
#' image(matrix(toyVideo[,10], ncol=100, nrow=100)) # frame
#' image(matrix(out$L[,10], ncol=100, nrow=100)) # seperated background
#' image(matrix(out$S[,10], ncol=100, nrow=100)) #seperated foreground
#' @export
rrpca <- function(A, lambda=NULL, maxiter=50, tol=1.0e-5, p=10, q=2, trace=FALSE, rand=TRUE) UseMethod("rrpca")
#' @export
rrpca.default <- function(A, lambda=NULL, maxiter=50, tol=1.0e-5, p=10, q=2, trace=FALSE, rand=TRUE) {
#*************************************************************************
#*** Author: N. Benjamin Erichson <nbe@st-andrews.ac.uk> ***
#*** <2016> ***
#*** License: BSD 3 clause ***
#*************************************************************************
A <- as.matrix(A)
m <- nrow(A)
n <- ncol(A)
rrpcaObj = list(L = NULL,
S = NULL,
err = NULL)
# Set target rank
k <- 1
if(k > min(m,n)) rrpcaObj$k <- min(m,n)
# Deal with missing values
is.na(A) <- 0
# Set lambda, gamma, rho
if(is.null(lambda)) lambda <- max(m,n)**-0.5
gamma <- 1.25
rho <- 1.5
if(rand == TRUE) {
svdalg = 'rsvd'
}else {
svdalg = 'svd'
}
# Compute matrix norms
spectralNorm <- switch(svdalg,
svd = norm(A, "2"),
rsvd = rsvd(A, k=1, p=10, q=1, nu=0, nv=0)$d,
stop("Selected SVD algorithm is not supported!")
)
infNorm <- norm( A , "I") / lambda
dualNorm <- max( spectralNorm , infNorm)
froNorm <- norm( A , "F")
# Initalize Lagrange multiplier
Z <- A / dualNorm
# Initialize tuning parameter
mu <- gamma / spectralNorm
mubar <- mu * 1e7
mu <- min( mu * rho , mubar )
muinv <- 1 / mu
# Init low-rank and sparse matrix
L = matrix(0, nrow = m, ncol = n)
S = matrix(0, nrow = m, ncol = n)
niter <- 1
err <- 1
while(err > tol && niter <= maxiter) {
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Update S using soft-threshold
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
epsi = lambda / mu
temp_S = A - L + Z / mu
S = matrix(0, nrow = m, ncol = n)
idxL <- which(temp_S < -epsi)
idxH <- which(temp_S > epsi)
S[idxL] <- temp_S[idxL] + epsi
S[idxH] <- temp_S[idxH] - epsi
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Singular Value Decomposition
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
R <- A - S + Z / mu
if(svdalg == 'svd') svd_out <- svd(R)
if(svdalg == 'rsvd') {
if(k > min(m,n)/5 ) auto_svd = 'svd' else auto_svd = 'rsvd'
svd_out <- switch(auto_svd,
svd = svd(R),
rsvd = rsvd(R, k=k+10, p=p, q=q))
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Predict optimal rank and update
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
svp = sum(svd_out$d > 1/mu)
if(svp <= k){
k = min(svp + 1, n)
} else {
k = min(svp + round(0.05 * n), n)
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Truncate SVD and update L
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# rrpcaObj$L = svd_out$u[,1:rrpcaObj$k] %*% diag(svd_out$d[1:rrpcaObj$k] - muinv, nrow=rrpcaObj$k, ncol=rrpcaObj$k) %*% t(svd_out$v[,1:rrpcaObj$k])
L = t(t(svd_out$u[,1:svp, drop=FALSE]) * (svd_out$d[1:svp] - 1/mu)) %*% t(svd_out$v[,1:svp, drop=FALSE])
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Compute error
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Astar = A - L - S
Z = Z + Astar * mu
err = norm( Astar , 'F') / froNorm
rrpcaObj$err <- c(rrpcaObj$err, err)
if(trace==TRUE){
cat('\n', paste0('Iteration: ', niter ), paste0(' predicted rank = ', svp ), paste0(' target rank k = ', k ), paste0(' Fro. error = ', rrpcaObj$err[niter] ))
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Update mu
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
mu = min(mu * rho, mubar);
muinv = 1 / mu
niter = niter + 1
}# End while loop
rrpcaObj$L <- L
rrpcaObj$S <- S
class(rrpcaObj) <- "rrpca"
return( rrpcaObj )
}
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.