# riem.wasserstein: Wasserstein Distance between Empirical Measures In Riemann: Learning with Data on Riemannian Manifolds

## 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

 1 2 3 4 5 6 7 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 weight1a length-M weight vector for μ; if NULL (default), uniform weight is set. weight2a 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 #------------------------------------------------------------------- # 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 June 20, 2021, 5:07 p.m.