Riemann (version 0.1.6)

wrap.stiefel: Prepare Data on (Compact) Stiefel Manifold

Description

Stiefel manifold \(St(k,p)\) is the set of \(k\)-frames in \(\mathbf{R}^p\), which is indeed a Riemannian manifold. For usage in Riemann package, each data point is represented as a matrix by the convention $$St(k,p) = \lbrace X \in \mathbf{R}^{p\times k} ~\vert~ X^\top X = I_k \rbrace$$ which means that columns are orthonormal. When the provided matrix is not an orthonormal basis as above, wrap.stiefel applies orthogonalization to extract valid basis information.

Usage

wrap.stiefel(input)

Value

a named riemdata S3 object containing

data

a list of \(k\)-frame orthonormal matrices.

size

size of each \(k\)-frame basis matrix.

name

name of the manifold of interests, "stiefel"

Arguments

input

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

array

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

list

a length-\(n\) list whose elements are \((p\times k)\) \(k\)-frames.

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations in St(2,4)
#-------------------------------------------------------------------
#  Data Generation by QR Decomposition
d1 = array(0,c(4,2,5))
d2 = list()
for (i in 1:5){
  d1[,,i] = qr.Q(qr(matrix(rnorm(4*2),ncol=2)))
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.stiefel(d1)
test2 = wrap.stiefel(d2)