wrap.grassmann {Riemann}

Prepare Data on Grassmann Manifold

Description

Grassmann manifold Gr(k,p) is the set of k-planes, or k-dimensional subspaces in R^p, which means that for a given matrix Y \in \mathbf{R}{p\times k}, the column space SPAN(Y) is an element in Grassmann manifold. We use a convention that each element in Gr(k,p) is represented as an orthonormal basis (ONB) X \in \mathbf{R}^{p\times k} where

X^\top X = I_k.

If not provided in such a form, this wrapper takes a QR decomposition of the given data to recover a corresponding ONB.

Usage

wrap.grassmann(input)

Arguments

input

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

array

an (p\times k\times n) array where each slice along 3rd dimension is a k-subspace basis in dimension p.

list

a length-n list whose elements are (p\times k) basis for k-subspace.

Value

a named riemdata S3 object containing

data

a list of k-subspace basis matrices.

size

size of each k-subspace basis matrix.

name

name of the manifold of interests, "grassmann"

Examples

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

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