bary14C  R Documentation 
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.
Algorithmically, it is a subgradient method where the each subgradient is
approximated using the entropic regularization.
bary14C(
support,
atoms,
marginals = NULL,
weights = NULL,
lambda = 0.1,
p = 2,
...
)
bary14Cdist(
distances,
marginals = NULL,
weights = NULL,
lambda = 0.1,
p = 2,
...
)
support 
an 
atoms 
a length 
marginals 
marginal distribution for empirical measures; if 
weights 
weights for each individual measure; if 
lambda 
regularization parameter (default: 0.1). 
p 
an exponent for the order of the distance (default: 2). 
... 
extra parameters including

distances 
a length 
a lengthN
vector of probability vector.
cuturi_fast_2014T4transport
#
# 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 = 100
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 = bary14C(support, myatoms, lambda=0.5, maxiter=10)
comp2 = bary14C(support, myatoms, lambda=1, maxiter=10)
comp3 = bary14C(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)
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