When \((p\times p)\) positive semidefinite 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
decomposes matrices whose numerical rank is exactly \(k\). Higher-rank inputs
are rejected; rank reduction must be performed explicitly before wrapping.
Numerical rank uses eigenvalues exceeding \(64p\) times machine precision
relative to the largest absolute eigenvalue. Negative eigenvalues beyond this
roundoff tolerance are rejected.
Arguments
- input
data matrices to be wrapped as
riemdataclass. 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 positive semidefinite matrices, an integer in \([1,p-1]\).
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*2),nrow=10)
d1[,,i] = tcrossprod(dat)
d2[[i]] = d1[,,i]
}
# Run
test1 = wrap.spdk(d1, k=2)
test2 = wrap.spdk(d2, k=2)