Riemannian Manifold Metric Learning — riem.rmml • Riemann

Given \(N\) observations \(X_1, X_2, \ldots, X_N \in \mathcal{M}\) and corresponding label information, riem.rmml computes pairwise distance of data under Riemannian Manifold Metric Learning (RMML) framework based on equivariant embedding. When the number of data points is not sufficient, an inverse of scatter matrix does not exist analytically so the small regularization parameter \(\lambda\) is recommended with default value of \(\lambda=0.1\).

Usage

riem.rmml(riemobj, label, lambda = 0.1, as.dist = FALSE)

Arguments

riemobj: a S3 "riemdata" class for \(N\) manifold-valued data.
label: a length-\(N\) vector of class labels. NA values are omitted.
lambda: regularization parameter. If \(\lambda \leq 0\), no regularization is applied.
as.dist: logical; if TRUE, it returns dist object, else it returns a symmetric matrix.

Value

a S3 dist object or \((N\times N)\) symmetric matrix of pairwise distances according to as.dist parameter.

References

Zhu P, Cheng H, Hu Q, Wang Q, Zhang C (2018). “Towards Generalized and Efficient Metric Learning on Riemannian Manifold.” In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, 3235--3241. ISBN 978-0-9992411-2-7.

Examples

#-------------------------------------------------------------------
#            Distance between Two Classes of SPD Matrices
#
#  Class 1 : Empirical Covariance from Standard Normal Distribution
#  Class 2 : Empirical Covariance from Perturbed 'iris' dataset
#-------------------------------------------------------------------
## DATA GENERATION
data(iris)
ndata  = 10
mydata = list()
for (i in 1:ndata){
  mydata[[i]] = stats::cov(matrix(rnorm(100*4),ncol=4))
}
for (i in (ndata+1):(2*ndata)){
  tmpdata = as.matrix(iris[,1:4]) + matrix(rnorm(150*4,sd=0.5),ncol=4)
  mydata[[i]] = stats::cov(tmpdata)
}
myriem = wrap.spd(mydata)
mylabs = rep(c(1,2), each=ndata)

## COMPUTE GEODESIC AND RMML PAIRWISE DISTANCE
pdgeo = riem.pdist(myriem)
pdmdl = riem.rmml(myriem, label=mylabs)

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(pdgeo[,(2*ndata):1], main="geodesic distance", axes=FALSE)
image(pdmdl[,(2*ndata):1], main="RMML distance", axes=FALSE)

par(opar)