rrpca: Randomized robust principal component analysis (rrpca).

In rsvd: Randomized Singular Value Decomposition

Description

Robust principal components analysis separates a matrix into a low-rank plus sparse component.

Usage

rrpca(
  A,
  lambda = NULL,
  maxiter = 50,
  tol = 1e-05,
  p = 10,
  q = 2,
  trace = FALSE,
  rand = TRUE
)

Arguments

A	array_like; a real (m, n) input matrix (or data frame) to be decomposed. na.omit is applied, if the data contain NAs.
lambda	scalar, optional; tuning parameter (default lambda = max(m,n)^-0.5).
maxiter	integer, optional; maximum number of iterations (default maxiter = 50).
tol	scalar, optional; precision parameter (default tol = 1.0e-5).
p	integer, optional; oversampling parameter for rsvd (default p=10), see rsvd.
q	integer, optional; number of additional power iterations for rsvd (default q=2), see rsvd.
trace	bool, optional; print progress.
rand	bool, optional; if (TRUE), the rsvd routine is used, otherwise svd is used.

Details

Robust principal component analysis (RPCA) is a method for the robust seperation of a a rectangular (m,n) matrix A into a low-rank component L and a sparse comonent S:

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.

Value

rrpca returns a list containing the following components:

L

array_like;
low-rank component; (m, n) dimensional array.

S

array_like
sparse component; (m, n) dimensional array.

Author(s)

N. Benjamin Erichson, [e-mail]

References

• [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.

• [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 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

Example output

 Iteration: 1      k = 2       Fro. error = 0.737746365366033
 Iteration: 2      k = 2       Fro. error = 0.192062179612869
 Iteration: 3      k = 3       Fro. error = 0.202330435689014
 Iteration: 4      k = 3       Fro. error = 0.285855997379785
 Iteration: 5      k = 3       Fro. error = 0.151270122099391
 Iteration: 6      k = 2       Fro. error = 0.0608696315891474
 Iteration: 7      k = 2       Fro. error = 0.0153711242624809
 Iteration: 8      k = 2       Fro. error = 0.0028854840421088
 Iteration: 9      k = 3       Fro. error = 0.00861708222513708
 Iteration: 10      k = 2       Fro. error = 0.00656480391276222
 Iteration: 11      k = 2       Fro. error = 0.00157598928211446
 Iteration: 12      k = 3       Fro. error = 0.00133994636922928
 Iteration: 13      k = 2       Fro. error = 0.000605321450071603

rsvd documentation built on April 16, 2021, 9:06 a.m.