riem.wasserstein: Wasserstein Distance between Empirical Measures

In Riemann: Learning with Data on Riemannian Manifolds

Wasserstein Distance between Empirical Measures

Description

Given two empirical measures μ, ν consisting of M and N observations, p-Wasserstein distance for p≥q 1 between two empirical measures is defined as

\mathcal{W}_p (μ, ν) = ≤ft( \inf_{γ \in Γ(μ, ν)} \int_{\mathcal{M}\times \mathcal{M}} d(x,y)^p d γ(x,y) \right)^{1/p}

where Γ(μ, ν) denotes the collection of all measures/couplings on \mathcal{M}\times \mathcal{M} whose marginals are μ and ν on the first and second factors, respectively.

Usage

riem.wasserstein(
  riemobj1,
  riemobj2,
  p = 2,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

riemobj1	a S3 "riemdata" class for M manifold-valued data, which are atoms of μ.
riemobj2	a S3 "riemdata" class for N manifold-valued data, which are atoms of ν.
p	an exponent for Wasserstein distance \mathcal{W}_p (default: 2).
geometry	(case-insensitive) name of geometry; either geodesic ("intrinsic") or embedded ("extrinsic") geometry.
...	extra parameters including weight1 a length-M weight vector for μ; if NULL (default), uniform weight is set. weight2 a length-N weight vector for ν; if NULL (default), uniform weight is set.

Value

a named list containing

distance

\mathcal{W_p} distance between two empirical measures.

plan

an (M\times N) matrix whose rowSums and columnSums are weight1 and weight2 respectively.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
# class 1 : 20 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 30 perturbed data points near (0,1,0) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata1 = list()
mydata2 = list()
for (i in 1:20){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata1[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 1:30){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem1 = wrap.sphere(mydata1)
myriem2 = wrap.sphere(mydata2)

## COMPUTE p-WASSERSTEIN DISTANCES
dist1 = riem.wasserstein(myriem1, myriem2, p=1)
dist2 = riem.wasserstein(myriem1, myriem2, p=2)
dist5 = riem.wasserstein(myriem1, myriem2, p=5)

pm1 = paste0("p=1: dist=",round(dist1$distance,3))
pm2 = paste0("p=2: dist=",round(dist2$distance,3))
pm5 = paste0("p=5: dist=",round(dist5$distance,3))

## VISUALIZE TRANSPORT PLAN AND DISTANCE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
image(dist1$plan, axes=FALSE, main=pm1)
image(dist2$plan, axes=FALSE, main=pm2)
image(dist5$plan, axes=FALSE, main=pm5)
par(opar)

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