swdist: Sliced Wasserstein Distance

In T4transport: Tools for Computational Optimal Transport

Sliced Wasserstein Distance

Description

Sliced Wasserstein (SW) Distance \insertCiterabin_2012_WassersteinBarycenterItsT4transport is a popular alternative to the standard Wasserstein distance due to its computational efficiency on top of nice theoretical properties. For the d-dimensional probability measures \mu and \nu, the SW distance is defined as

\mathcal{SW}_p (\mu, \nu) = \left( \int_{\mathbf{S}^{d-1}} \mathcal{W}_p^p ( \langle \theta, \mu\rangle, \langle \theta, \nu \rangle d\lambda (\theta) \right)^{1/p},

where \mathbf{S}^{d-1} is the (d-1)-dimensional unit hypersphere and \lambda is the uniform distribution on \mathbf{S}^{d-1}. Practically, it is computed via Monte Carlo integration.

Usage

swdist(X, Y, p = 2, ...)

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).
...	extra parameters including nproj the number of Monte Carlo samples for SW computation (default: 496).

Value

a named list containing

distance

\mathcal{SW}_p distance value.

projdist

a length-niter vector of projected univariate distances.

References

\insertAllCited

Examples

#-------------------------------------------------------------------
#  Sliced-Wasserstein Distance between Two Bivariate Normal
#
# * class 1 : samples from Gaussian with mean=(-1, -1)
# * class 2 : samples from Gaussian with mean=(+1, +1)
#-------------------------------------------------------------------
# SMALL EXAMPLE
set.seed(100)
m = 20
n = 30
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 THE SLICED-WASSERSTEIN DISTANCE
outsw <- swdist(X, Y, nproj=100)

# VISUALIZE
# prepare ingredients for plotting
plot_x = 1:1000
plot_y = base::cumsum(outsw$projdist)/plot_x

# draw
opar <- par(no.readonly=TRUE)
plot(plot_x, plot_y, type="b", cex=0.1, lwd=2,
     xlab="number of MC samples", ylab="distance",
     main="Effect of MC Sample Size")
abline(h=outsw$distance, col="red", lwd=2)
legend("bottomright", legend="SW Distance", 
       col="red", lwd=2)
par(opar)

T4transport documentation built on April 12, 2023, 12:37 p.m.