# wrap.stiefel: Prepare Data on (Compact) Stiefel Manifold In Riemann: Learning with Data on Riemannian Manifolds

 wrap.stiefel R Documentation

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

#-------------------------------------------------------------------
#                 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)



Riemann documentation built on March 18, 2022, 7:55 p.m.