# bary15B: Barycenter by Benamou et al. (2015) In T4transport: Tools for Computational Optimal Transport

 bary15B R Documentation

## Barycenter by Benamou et al. (2015)

### Description

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 abstolstopping criterion for iterations (default: 1e-10). init.vecan initial vector (default: uniform weight). maxitermaximum number of iterations (default: 496). print.progressa 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

\insertRef

benamou_iterative_2015T4transport

### 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