gwbary: Gromov-Wasserstein Barycenter

In T4transport: Tools for Computational Optimal Transport

Gromov-Wasserstein Barycenter

Description

Computes the Gromov–Wasserstein (GW) barycenter of a collection of metric measure spaces. Given a list of distance matrices {D^{(k)}}{k=1}^K and their corresponding marginal distributions, the function estimates a synthetic metric space whose intrinsic geometry best represents the input collection under the GW criterion.

The GW barycenter is defined as the minimizer of a multi-measure Gromov–Wasserstein objective, where each dataset contributes according to a user-specified barycentric weight. Since the problem is jointly non-convex in the barycenter metric and the coupling matrices, the algorithm proceeds through an outer–inner iterative procedure.

Usage

gwbary(distances, marginals = NULL, weights = NULL, num_support = 100, ...)

Arguments

distances	a length-K list where each element is either an (N_k \times N_k) distance matrix or an object of class dist representing the pairwise distances for each empirical measure.
marginals	marginal distributions for empirical measures; if NULL (default), uniform weights are set for all measures. Otherwise, it should be a length-K list where each element is a length-N_k vector of nonnegative weights that sum to 1.
weights	weights for each individual measure; if NULL (default), each measure is considered equally. Otherwise, it should be a length-K vector.
num_support	the number of support points M for the barycenter (default: 100).
...	extra parameters including maxiter maximum number of iterations (default: 10). abstol stopping criterion for iterations (default: 1e-6). method optimization method to use; can be one of "mm", "pg", or "fw" (default).

Value

A named list containing

dist

an object of class dist representing the GW barycenter.

weight

a length-M vector of barycenter weights with all entries being 1/M.

Examples

## Not run: 
#-------------------------------------------------------------------
#                           Description
#
# GW barycenter computation is quite expensive. In this example, 
# we draw a small set of empirical measures from the digit '3'
# images and compute their GW barycenter with a small number of 
# support points. The attained barycenter distance matrix is then 
# passed onto the classical MDS algorithm for visualization.
#-------------------------------------------------------------------
## GENERATE DATA
data(digits)
data_D = vector("list", length=5)
data_W = vector("list", length=5)
for (i in 1:5){
  img_now = img2measure(digits3[[i]])
  data_D[[i]] = stats::dist(img_now$support)
  data_W[[i]] = as.vector(img_now$weight)
}

## COMPUTE
bary_dist <- gwbary(data_D, marginals=data_W, num_support=100)
bary_cmd2 <- stats::cmdscale(bary_dist$dist, k=2)

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(pty="s")
plot(bary_cmd2, main="GW Barycenter Embedding",
     xaxt="n", yaxt="n", pch=19, xlab="", ylab="")
par(opar)

## End(Not run)

T4transport documentation built on Jan. 11, 2026, 9:07 a.m.