ALS.CPC {ALSCPC}

minimize the objective function \Phi(\bold{D}) by using of the accelerated line search algorithm

Description

The ALS.CPC function implement ALS algorithm based on the update formula

\bold{D_{k+1}} = R_{\bold{D_k}} (-\beta^{m_k} \ \alpha \ grad(\Phi (\bold{D}_k)))

until convergence (i.e. |\Phi(\bold{D}_k)-\Phi (\bold{D}_{k+1})| \le \epsilon) and return the orthogonal matrix \bold{D}_r, r is the smallest nonnegative integer k such that |\Phi(\bold{D}_k)-\Phi (\bold{D}_{k+1})| \le \epsilon.

Usage

ALS.CPC(alpha,beta,sigma,epsilon,G,nval,D,S)

Arguments

alpha

positive real number.

beta

real number belong to (0,1).

sigma

real number belong to (0,1).

epsilon

small positive constant controlling error term.

G

number of groups in common principal components analysis.

nval

a numeric vector containing the positive integers of sample sizes minus one in each group.

D

an initial square orthogonal matrix of order p, where p is group dimensionality.

S

a list of length G of positive definite symmetric matrices of order p.

Value

An orthogonal matrix such that minimize \Phi(\bold{D}).

Author(s)

Dariush Najarzadeh

References

Absil, P. A., Mahony, R., & Sepulchre, R. (2009). Optimization algorithms on matrix manifolds. Princeton University Press.

Examples

nval<-numeric(3) 
nval[[1]]<-49
nval[[2]]<-49
nval[[3]]<-49
S<-vector("list",length=3)
setosa<-iris[1:50,1:4]; names(setosa)<-NULL
versicolor<-iris[51:100,1:4]; names(versicolor)<-NULL
virginica<-iris[101:150,1:4]; names(virginica)<-NULL
S[[1]]<-as.matrix(var(versicolor))
S[[2]]<-as.matrix(var(virginica))
S[[3]]<-as.matrix(var(setosa))
D<-diag(4)
ALS.CPC(10,0.5,0.4,1e-5,G=3,nval,D,S)