Riemann (version 0.1.4)

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)

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.

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"

Examples

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

# }