# R/rrpca.R In Benli11/rPCA: Randomized Singular Value Decomposition

#### Documented in rrpca

#' @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:
#'    \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{[email protected]}
#'
#' @references
#' \itemize{
#'   \item  [1] 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{http://arxiv.org/abs/1009.5055}).
#'
#'   \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}).
#' }
#'
#' @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)
#'
#' 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 <[email protected]>        ***
#***                              <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 )

}
`
Benli11/rPCA documentation built on Nov. 6, 2018, 10:46 p.m.