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.
Arguments
- input
data matrices to be wrapped as
riemdataclass. 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)