st.pdist: Pairwise Distance for Data on Stiefel Manifold

In RiemStiefel: Inference, Learning, and Optimization on Stiefel Manifold

Description

For data on Stiefel manifold x_1,x_2,…,x_N \in St(r,p), compute pairwise distances d(x_i,x_j) via geodesic ("intrinsic") distance or embedded Euclidean "extrinsic" metric. Since computing geodesic has no closed-form expressions, it relies on a numerical approximation which may incur heavier computational burden.

Usage

st.pdist(x, type = c("intrinsic", "extrinsic"), as.dist = TRUE)

Arguments

x	either an array of size (p\times r\times N) or a list of length N whose elements are (p\times r) matrix on Stiefel manifold.
type	type of distance, either "intrinsic" or "extrinsic".
as.dist	a logical; TRUE to return a dist object or FALSE to return an (N\times N) symmetric matrix.

Value

a dist object or (N\times N) symmetric matrix depending on as.dist.

Author(s)

Kisung You

Examples

#-------------------------------------------------------------------
#      Generate a dataset with two types of Stiefel elements
#-------------------------------------------------------------------
#  group1 : first four columns of (8x8) identity matrix + noise
#  group2 : last  four columns of (8x8) identity matrix + noise

mydata = list()
sdval  = 0.05
diag8  = diag(8)
for (i in 1:10){
  mydata[[i]] = qr.Q(qr(diag8[,1:4] + matrix(rnorm(8*4,sd=sdval),ncol=4)))
}
for (i in 11:20){
  mydata[[i]] = qr.Q(qr(diag8[,5:8] + matrix(rnorm(8*4,sd=sdval),ncol=4)))
}

## compare 'intrinsic' and 'extrinsic' distances
dint = st.pdist(mydata, type="intrinsic", as.dist=FALSE)
dext = st.pdist(mydata, type="extrinsic", as.dist=FALSE)

## visualize
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(dint[,20:1], main="intrinsic")
image(dext[,20:1], main="extrinsic")
par(opar)

RiemStiefel documentation built on March 26, 2020, 7:48 p.m.