barycenter_lp: Exact computation of 2-Wasserstein barycenters in R^d using linear programming

Description

This function solves the 2-Wasserstein barycenter problem of N finitely supported input measures explicitly by solving the corresponding linear program.

Usage

barycenter_lp(pos.list, weights.list, frechet.weights = NULL)

Arguments

pos.list

A list of Mxd matrices, specifying the positions of the data measures.

weights.list

A list of vectors with non-negative entries and identical total sum specifying the weights of the data measures.

frechet.weights

A vector of positive entries summing to one specifying the weights of each data measure in the Fr<U+00E9>chet functional.

Value

A list with two entries. The first entry contains the positions of the computed barycenter and the second entry contains the corresponding weights.

References

E Anderes, S Borgwardt, and J Miller (2016). Discrete Wasserstein barycenters: Optimal transport for discrete data. Mathematical Methods of Operations Research, 84(2):389-409. S Borgwardt and S Patterson (2020). Improved linear programs for discrete barycenters. Informs Journal on Optimization 2(1):14-33.

Examples

Run this code

# NOT RUN {
pos.list<-vector("list",4)
weights.list<-vector("list",4)
pos.list[[1]]<-matrix(c(0,0,1,1,1,0,0,1),nrow=4,ncol=2)/10
pos.list[[2]]<-matrix(c(9,9,10,10,10,9,9,10),nrow=4,ncol=2)/10
pos.list[[3]]<-matrix(c(9,9,10,10,1,0,0,1),nrow=4,ncol=2)/10
pos.list[[4]]<-matrix(c(0,0,1,1,10,9,9,10),nrow=4,ncol=2)/10
plot(0, 0, xlab = "", ylab = "", type = "n", xlim = c(0, 1), ylim = c(0, 1))
for(i in 1:4)
  points(pos.list[[i]][,1], pos.list[[i]][,2], col = i)
weights.list[[1]]<-rep(1/4,4)
weights.list[[2]]<-rep(1/4,4)
weights.list[[3]]<-rep(1/4,4)
weights.list[[4]]<-rep(1/4,4)
bary<-barycenter_lp(pos.list,weights.list)
points(bary$positions[,1],bary$positions[,2], col = "orange", pch = 13)
# }