Barycenter by Benamou et al. (2015)

Given \(K\) empirical measures \(\mu_1, \mu_2, \ldots, \mu_K\) of possibly different cardinalities, wasserstein barycenter \(\mu^*\) is the solution to the following problem $$\sum_{k=1}^K \pi_k \mathcal{W}_p^p (\mu, \mu_k)$$ where \(\pi_k\)'s are relative weights of empirical measures. Here we assume either (1) support atoms in Euclidean space are given, or (2) all pairwise distances between atoms of the fixed support and empirical measures are given. Authors proposed iterative Bregman projections in conjunction with entropic regularization.

Usage

bary15B(
  support,
  atoms,
  marginals = NULL,
  weights = NULL,
  lambda = 0.1,
  p = 2,
  ...
)

bary15Bdist(
  distances,
  marginals = NULL,
  weights = NULL,
  lambda = 0.1,
  p = 2,
  ...
)

Arguments

support

an \((N\times P)\) matrix of rows being atoms for the fixed support.

atoms

a length-\(K\) list where each element is an \((N_k \times P)\) matrix of atoms.

marginals

marginal distribution 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-\(N_i\) 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.

lambda

regularization parameter (default: 0.1).

p

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

...

extra parameters including

abstol

stopping criterion for iterations (default: 1e-10).

init.vec

an initial vector (default: uniform weight).

maxiter

maximum number of iterations (default: 496).

print.progress

a logical to show current iteration (default: FALSE).

distances

a length-\(K\) list where each element is an \((N\times N_k)\) pairwise distance between atoms of the fixed support and given measures.

Value

a length-\(N\) vector of probability vector.

References

Benamou J, Carlier G, Cuturi M, Nenna L, Peyré G (2015). “Iterative Bregman Projections for Regularized Transportation Problems.” SIAM Journal on Scientific Computing, 37(2), A1111--A1138. ISSN 1064-8275, 1095-7197.

Examples

#-------------------------------------------------------------------
#     Wasserstein Barycenter for Fixed Atoms with Two Gaussians
#
# * class 1 : samples from Gaussian with mean=(-4, -4)
# * class 2 : samples from Gaussian with mean=(+4, +4)
# * target support consists of 7 integer points from -6 to 6,
#   where ideally, weight is concentrated near 0 since it's average!
#-------------------------------------------------------------------
## GENERATE DATA
#  Empirical Measures
set.seed(100)
ndat = 500
dat1 = matrix(rnorm(ndat*2, mean=-4, sd=0.5),ncol=2)
dat2 = matrix(rnorm(ndat*2, mean=+4, sd=0.5),ncol=2) 

myatoms = list()
myatoms[[1]] = dat1
myatoms[[2]] = dat2
mydata = rbind(dat1, dat2)

#  Fixed Support
support = cbind(seq(from=-8,to=8,by=2),
                seq(from=-8,to=8,by=2))
## COMPUTE
comp1 = bary15B(support, myatoms, lambda=0.5, maxiter=10)
comp2 = bary15B(support, myatoms, lambda=1,   maxiter=10)
comp3 = bary15B(support, myatoms, lambda=5,   maxiter=10)

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
barplot(comp1, main="lambda=0.5")
barplot(comp2, main="lambda=1")
barplot(comp3, main="lambda=5")

par(opar)