Barycenter of Histograms by Benamou et al. (2015)

Given multiple histograms represented as "histogram" S3 objects, compute Wasserstein barycenter. We need one requirement that all histograms in an input list hists must have same breaks. See the example on how to construct a histogram on predefined breaks/bins.

Usage

histbary15B(hists, p = 2, weights = NULL, lambda = NULL, ...)

Arguments

hists

a length-\(N\) list of histograms ("histogram" object) of same breaks.

p

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

weights

a weight of each image; if NULL (default), uniform weight is set. Otherwise, it should be a length-\(N\) vector of nonnegative weights.

lambda

a regularization parameter; if NULL (default), a paper's suggestion would be taken, or it should be a nonnegative real number.

...

extra parameters including

abstol: stopping criterion for iterations (default: 1e-8).
init.vec: an initial weight vector (default: uniform weight).
maxiter: maximum number of iterations (default: 496).
nthread: number of threads for OpenMP run (default: 1).
print.progress: a logical to show current iteration (default: TRUE).

Value

a "histogram" object of barycenter.

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

#----------------------------------------------------------------------
#                      Binned from Two Gaussians
#
# EXAMPLE : Very Small Example for CRAN; just showing how to use it!
#----------------------------------------------------------------------
# GENERATE FROM TWO GAUSSIANS WITH DIFFERENT MEANS
set.seed(100)
x  = stats::rnorm(1000, mean=-4, sd=0.5)
y  = stats::rnorm(1000, mean=+4, sd=0.5)
bk = seq(from=-10, to=10, length.out=20)

# HISTOGRAMS WITH COMMON BREAKS
histxy = list()
histxy[[1]] = hist(x, breaks=bk, plot=FALSE)
histxy[[2]] = hist(y, breaks=bk, plot=FALSE)

# COMPUTE
hh = histbary15B(histxy, maxiter=5)

# VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
barplot(histxy[[1]]$density, col=rgb(0,0,1,1/4), 
        ylim=c(0, 0.75), main="Two Histograms")
barplot(histxy[[2]]$density, col=rgb(1,0,0,1/4), 
        ylim=c(0, 0.75), add=TRUE)
barplot(hh$density, main="Barycenter",
        ylim=c(0, 0.75))

par(opar)