SinkhornDistance: Sinkhorn Distances (upper bound to EMD)

In Barycenter: Regularized Wasserstein Distances and Barycenters

Description

The Sinkhorn algorithm to compute N dual-Sinkhorn divergences, i.e. upper bounds on the earth movers distance (EMD), a.k.a. Wasserstein distance.

Usage

Sinkhorn(a, b, costm, lambda = 1, maxIter = 10000, tolerance=10^(-8))

Arguments

a	Either a d_1 x 1 column vector in the probability simplex (nonnegative summing to one) or a d_1 x N matrix, where each column is in the probability simplex.
b	A d_1 x N matrix of N vectors in the probability simplex.
costm	A matrix of pairwise distances/costs between bins described in a and bins in the b_1,...,b_N histograms.
lambda	Non-negative regularization parameter (for small lambda the Sinkhorn Distance is close to the EMD).
maxIter	Maximum number of iterations.
tolerance	A threshold for the integrated stopping criterion based on marginal differences.

Value

Returns the Sinkhorn Distances between the given bins. If a is given by a d_1 x 1 column vector the function returns the distances

[D(a,b_1),...,D(a,b_N)].

If a is given by a d_1 x N matrix the function returns the distances

[D(a_1,b_1),...,D(a_N,b_N)].

If a and b are simply given by two d_1 x 1 and d_2 x 1 vectors each in the probability simplex, respectively, then the functions returns a list containing the Sinkhorn Distance as well as the corresponding regularized transport plan.

Author(s)

Marcel Klatt

References

Cuturi, M.: Sinkhorn Distances: Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems 26, 2013

Examples

#Sinkhorn Distances between the first image to the remaining four images in the dataset eight.
#We creat costm simply using a distance matrix on the grid [0,1]x[0,1].
n <- seq(0,1,length.out = dim(eight[[1]])[2])
costm <- as.matrix(dist(expand.grid(n,rev(n)), diag=TRUE, upper=TRUE))
a <- matrix(eight[[1]],28*28,1)
b <- matrix(c(eight[[2]],eight[[3]],eight[[4]],eight[[5]]),28*28,4)
Sinkhorn(a, b, costm)

Example output

Loading required package: Rcpp
$Distance
          [,1]      [,2]      [,3]      [,4]
[1,] 0.2945937 0.3057875 0.3189765 0.3020667

Barycenter documentation built on May 2, 2019, 6:47 a.m.