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)

if (FALSE) {
## 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