wasserstein {T4transport}

Wasserstein Distance between Empirical Measures

Description

Given two empirical measures \mu, \nu consisting of M and N observations on \mathcal{X}, p-Wasserstein distance for p\geq 1 between two empirical measures is defined as

\mathcal{W}_p (\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathcal{X}\times \mathcal{X}} d(x,y)^p d \gamma(x,y) \right)^{1/p}

where \Gamma(\mu, \nu) denotes the collection of all measures/couplings on \mathcal{X}\times \mathcal{X} whose marginals are \mu and \nu on the first and second factors, respectively. Please see the section for detailed description on the usage of the function.

Usage

wasserstein(X, Y, p = 2, wx = NULL, wy = NULL)

wassersteinD(D, p = 2, wx = NULL, wy = NULL)

Arguments

X

an (M\times P) matrix of row observations.

Y

an (N\times P) matrix of row observations.

p

an exponent for the order of the distance (default: 2).

wx

a length-M marginal density that sums to 1. If NULL (default), uniform weight is set.

wy

a length-N marginal density that sums to 1. If NULL (default), uniform weight is set.

D

an (M\times N) distance matrix d(x_m, y_n) between two sets of observations.

Value

a named list containing

distance

\mathcal{W}_p distance value.

plan

an (M\times N) nonnegative matrix for the optimal transport plan.

Using wasserstein() function

We assume empirical measures are defined on the Euclidean space \mathcal{X}=\mathbf{R}^d,

\mu = \sum_{m=1}^M \mu_m \delta_{X_m}\quad\textrm{and}\quad \nu = \sum_{n=1}^N \nu_n \delta_{Y_n}

and the distance metric used here is standard Euclidean norm d(x,y) = \|x-y\|. Here, the marginals (\mu_1,\mu_2,\ldots,\mu_M) and (\nu_1,\nu_2,\ldots,\nu_N) correspond to wx and wy, respectively.

Using wassersteinD() function

If other distance measures or underlying spaces are one's interests, we have an option for users to provide a distance matrix D rather than vectors, where

D := D_{M\times N} = d(X_m, Y_n)

for flexible modeling.

References

Peyré G, Cuturi M (2019). “Computational Optimal Transport: With Applications to Data Science.” Foundations and Trends® in Machine Learning, 11(5-6), 355–607. ISSN 1935-8237, 1935-8245.

Examples

#-------------------------------------------------------------------
#  Wasserstein Distance between Samples from Two Bivariate Normal
#
# * class 1 : samples from Gaussian with mean=(-1, -1)
# * class 2 : samples from Gaussian with mean=(+1, +1)
#-------------------------------------------------------------------
## SMALL EXAMPLE
m = 20
n = 10
X = matrix(rnorm(m*2, mean=-1),ncol=2) # m obs. for X
Y = matrix(rnorm(n*2, mean=+1),ncol=2) # n obs. for Y

## COMPUTE WITH DIFFERENT ORDERS
out1 = wasserstein(X, Y, p=1)
out2 = wasserstein(X, Y, p=2)
out5 = wasserstein(X, Y, p=5)

## VISUALIZE : SHOW THE PLAN AND DISTANCE
pm1 = paste0("plan p=1; distance=",round(out1$distance,2))
pm2 = paste0("plan p=2; distance=",round(out2$distance,2))
pm5 = paste0("plan p=5; distance=",round(out5$distance,2))

opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
image(out1$plan, axes=FALSE, main=pm1)
image(out2$plan, axes=FALSE, main=pm2)
image(out5$plan, axes=FALSE, main=pm5)
par(opar)

## Not run: 
## COMPARE WITH ANALYTIC RESULTS
#  For two Gaussians with same covariance, their 
#  2-Wasserstein distance is known so let's compare !

niter = 1000          # number of iterations
vdist = rep(0,niter)
for (i in 1:niter){
  mm = sample(30:50, 1)
  nn = sample(30:50, 1)
  
  X = matrix(rnorm(mm*2, mean=-1),ncol=2)
  Y = matrix(rnorm(nn*2, mean=+1),ncol=2)
  
  vdist[i] = wasserstein(X, Y, p=2)$distance
  if (i%%10 == 0){
    print(paste0("iteration ",i,"/", niter," complete.")) 
  }
}

# Visualize
opar <- par(no.readonly=TRUE)
hist(vdist, main="Monte Carlo Simulation")
abline(v=sqrt(8), lwd=2, col="red")
par(opar)

## End(Not run)