Sample‐Based Optimal Transport and Barycenter Problems

Abstract

A methodology is developed for the numerical solution to the sample‐based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets. © 2019 Wiley Periodicals, Inc.

Adaptive optimal transport

“…(b) Replacing "all" functions g(x) by a suitable family of functions provides a natural relaxation in the presence of finite sample sets. We show that, unlike the maps T t , which produce the global map T via composition, it is the sum of the functions g t that approximates the global g. Moreover, we prove that, if the families of T t and g t are built through the linear superposition of a predetermined set of functions, we recover the solution in [7].…”

“…In [7], a set of 'features' f 1 , .., f K serve as test functions to evaluate the statement ρ = ν for ρ, ν ∈ P (R d ), when we only have sample points (z i ) i=1,..,n and (y j ) j=1,..,m generated from Z ∼ ρ, Y ∼ ν.…”

“…1. Borrowing from the methodology developed in [7], we subdivide the transportation problem between µ and ν into finding N local maps T t pushing forward ρ t−1 to ρ t , with ρ 0 = µ and ρ N = ν. The global map T results from the composition of these local maps: T = T N • T N −1 • .…”

“…In [7], the push-forward condition was formulated in terms of the equality of the empirical expected values of a pre-determined set of feature functions. Instead, we propose a broader and adaptive formulation, in terms of the relative entropy between the two distributions.…”

“…We can assume that the source and target distribution are close:Remark. Solving the problem for nearby distributions is the building block of a general procedure for arbitrary distributions and for finding the Wasserstein barycenter of distributions [7]. This more general procedure is presented in Section 2.4.…”