wrap.spdk {Riemann}

Prepare Data on SPD Manifold of Fixed-Rank

Description

When (p\times p) SPD matrices are of fixed-rank k < p, they form a geometric structure represented by (p\times k) matrices,

SPD(k,p) = \lbrace X \in \mathbf{R}^{(p\times p)}~\vert~ Y Y^\top = X, \textrm{rank}(X) = k \rbrace

It's key difference from \mathcal{S}_{++}^p is that all matrices should be of fixed rank k where k is usually smaller than p. Inputs are given as (p\times p) matrices with specified k and wrap.spdk automatically decomposes input square matrices into rank-k representation matrices.

Usage

wrap.spdk(input, k)

Arguments

input

data matrices to be wrapped as riemdata class. Following inputs are considered,

array

a (p\times p\times n) array where each slice along 3rd dimension is a rank-k matrix.

list

a length-n list whose elements are (p\times p) matrices of rank-k.

k

rank of the SPD matrices.

Value

a named riemdata S3 object containing

data

a list of (p\times k) representation of the corresponding rank-k SPSD matrices.

size

size of each representation matrix.

name

name of the manifold of interests, "spdk"

References

Journée M, Bach F, Absil P, Sepulchre R (2010). “Low-rank optimization on the cone of positive semidefinite matrices.” SIAM Journal on Optimization, 20(5), 2327–2351.

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#-------------------------------------------------------------------
#  Data Generation
d1 = array(0,c(10,10,3))
d2 = list()
for (i in 1:3){
  dat = matrix(rnorm(10*10),ncol=10)
  d1[,,i] = stats::cov(dat)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.spdk(d1, k=2)
test2 = wrap.spdk(d2, k=2)