do.rpca {Rdimtools}

Robust Principal Component Analysis

Description

Robust PCA (RPCA) is not like other methods in this package as finding explicit low-dimensional embedding with reduced number of columns. Rather, it is more of a decomposition method of data matrix X, possibly noisy, into low-rank and sparse matrices by solving the following,

\textrm{minimize}\quad \|L\|_* + \lambda \|S\|_1 \quad{s.t.} L+S=X

where L is a low-rank matrix, S is a sparse matrix and \|\cdot\|_* denotes nuclear norm, i.e., sum of singular values. Therefore, it should be considered as preprocessing procedure of denoising. Note that after RPCA is applied, L should be used as kind of a new data matrix for any manifold learning scheme to be applied.

Usage

do.rpca(X, mu = 1, lambda = sqrt(1/(max(dim(X)))), ...)

Arguments

X

an (n\times p) matrix or whose rows are observations and columns represent independent variables.

mu

an augmented Lagrangian parameter

lambda

parameter for the sparsity term \|S\|_1. Default value is given accordingly to the referred paper.

...

extra parameters including

maxiter

maximum number of iterations (default: 100).

abstol

absolute tolerance stopping criterion (default: 1e-8).

Value

a named list containing

L

an (n\times p) low-rank matrix.

S

an (n\times p) sparse matrix.

algorithm

name of the algorithm.

Author(s)

Kisung You

References

Candès EJ, Li X, Ma Y, Wright J (2011). “Robust Principal Component Analysis?” Journal of the ACM, 58(3), 1–37.

Examples

## load iris data and add some noise
data(iris, package="Rdimtools")
set.seed(100)
subid = sample(1:150,50)
noise = 0.2
X = as.matrix(iris[subid,1:4])
X = X + matrix(noise*rnorm(length(X)), nrow=nrow(X))
lab = as.factor(iris[subid,5])

## try different regularization parameters
rpca1 = do.rpca(X, lambda=0.1)
rpca2 = do.rpca(X, lambda=1)
rpca3 = do.rpca(X, lambda=10)

## apply identical PCA methods
Y1 = do.pca(rpca1$L, ndim=2)$Y
Y2 = do.pca(rpca2$L, ndim=2)$Y
Y3 = do.pca(rpca3$L, ndim=2)$Y

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(Y1, pch=19, col=lab, main="RPCA+PCA::lambda=0.1")
plot(Y2, pch=19, col=lab, main="RPCA+PCA::lambda=1")
plot(Y3, pch=19, col=lab, main="RPCA+PCA::lambda=10")
par(opar)