Riemann (version 0.1.4)

wrap.spdk: 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

journee_lowrank_2010Riemann

Examples

# NOT RUN {
#-------------------------------------------------------------------
#                 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)


# }