LRSC {T4cluster}

Low-Rank Subspace Clustering

Description

Low-Rank Subspace Clustering (LRSC) constructs the connectivity of the data by solving

\textrm{min}_C \|C\|_*\quad\textrm{such that}\quad A=AC,~C=C^\top

for the uncorrupted data scenario where A is a column-stacked data matrix. In the current implementation, the first equality constraint for reconstructiveness of the data can be relaxed by solving

\textrm{min}_C \|C\|_* + \frac{\tau}{2} \|A-AC\|_F^2 \quad\textrm{such that}\quad C=C^\top

controlled by the regularization parameter \tau. If you are interested in enabling a more general class of the problem suggested by authors, please contact maintainer of the package.

Usage

LRSC(data, k = 2, type = c("relaxed", "exact"), tau = 1)

Arguments

data

an (n\times p) matrix of row-stacked observations.

k

the number of clusters (default: 2).

type

type of the problem to be solved.

tau

regularization parameter for relaxed-constraint problem.

Details

\textrm{min}_C \|C\|_*\quad\textrm{such that}\quad D=DC

for column-stacked data matrix D and \|\cdot \|_* is the nuclear norm which is relaxation of the rank condition. If you are interested in full implementation of the algorithm with sparse outliers and noise, please contact the maintainer.

Value

a named list of S3 class T4cluster containing

cluster

a length-n vector of class labels (from 1:k).

algorithm

name of the algorithm.

References

Vidal R, Favaro P (2014). “Low Rank Subspace Clustering (LRSC).” Pattern Recognition Letters, 43, 47–61. ISSN 01678655.

Examples

## generate a toy example
set.seed(10)
tester = genLP(n=100, nl=2, np=1, iso.var=0.1)
data   = tester$data
label  = tester$class

## do PCA for data reduction
proj = base::eigen(stats::cov(data))$vectors[,1:2]
dat2 = data%*%proj

## run LRSC algorithm with k=2,3,4 with relaxed/exact solvers
out2rel = LRSC(data, k=2, type="relaxed")
out3rel = LRSC(data, k=3, type="relaxed")
out4rel = LRSC(data, k=4, type="relaxed")

out2exc = LRSC(data, k=2, type="exact")
out3exc = LRSC(data, k=3, type="exact")
out4exc = LRSC(data, k=4, type="exact")

## extract label information
lab2rel = out2rel$cluster
lab3rel = out3rel$cluster
lab4rel = out4rel$cluster

lab2exc = out2exc$cluster
lab3exc = out3exc$cluster
lab4exc = out4exc$cluster

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3))
plot(dat2, pch=19, cex=0.9, col=lab2rel, main="LRSC Relaxed:K=2")
plot(dat2, pch=19, cex=0.9, col=lab3rel, main="LRSC Relaxed:K=3")
plot(dat2, pch=19, cex=0.9, col=lab4rel, main="LRSC Relaxed:K=4")
plot(dat2, pch=19, cex=0.9, col=lab2exc, main="LRSC Exact:K=2")
plot(dat2, pch=19, cex=0.9, col=lab3exc, main="LRSC Exact:K=3")
plot(dat2, pch=19, cex=0.9, col=lab4exc, main="LRSC Exact:K=4")
par(opar)