gr.mean: Fréchet Mean on Grassmann Manifold

In RiemGrassmann: Inference, Learning, and Optimization on Grassmann Manifold

Description

For manifold-valued data, Fréchet mean is the solution of following cost function,

\textrm{min}_x ∑_{i=1}^n ρ^2 (x, x_i),\quad x\in\mathcal{M}

for a given data \{x_i\}_{i=1}^n and ρ(x,y) is the geodesic distance between two points on manifold \mathcal{M}. It uses a gradient descent method with a backtracking search rule for updating.

Usage

gr.mean(x, type = c("intrinsic", "extrinsic"), eps = 1e-06, parallel = FALSE)

Arguments

x	either an array of size (n\times k\times N) or a list of length N whose elements are (n\times k) orthonormal basis (ONB) on Grassmann manifold.
type	type of geometry, either "intrinsic" or "extrinsic".
eps	stopping criterion for the norm of gradient.
parallel	a flag for enabling parallel computation with OpenMP.

Value

a named list containing

mu

an estimated mean matrix for ONB of size (n\times k).

variation

Fréchet variation with the estimated mean.

Author(s)

Kisung You

Examples

## generate a dataset with two types of Grassmann elements
#  first four columns of (8x8) identity matrix + noise
mydata = list()
sdval  = 0.1
diag8  = diag(8)
for (i in 1:10){
  mydata[[i]] = qr.Q(qr(diag8[,1:4] + matrix(rnorm(8*4,sd=sdval),ncol=4)))
}

## compute two types of means
mean.int = gr.mean(mydata, type="intrinsic")
mean.ext = gr.mean(mydata, type="extrinsic")

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
image(mean.int$mu, main="intrinsic mean")
image(mean.ext$mu, main="extrinsic mean")
par(opar)

RiemGrassmann documentation built on March 25, 2020, 5:07 p.m.