Adaptive optimal transport

Abstract

An adaptive, adversarial methodology is developed for the optimal transport problem between two distributions µ and ν, known only through a finite set of independent samples (xi)i=1..N and (yj)j=1..M . The methodology automatically creates features that adapt to the data, thus avoiding reliance on a priori knowledge of data distribution. Specifically, instead of a discrete point-bypoint assignment, the new procedure seeks an optimal map T (x) defined for all x, minimizing the Kullback-Leibler divergence between (T (xi)) and the target (yj). The relative entropy is given a sample-based, variational characterization, thereby creating an adversarial setting: as one player seeks to push forward one distribution to the other, the second player develops features that focus on those areas where the two distributions fail to match. The procedure solves local problems matching consecutive, intermediate distributions between µ and ν. As a result, maps of arbitrary complexity can be built by composing the simple maps used for each local problem. Displaced interpolation is used to guarantee global from local optimality. The procedure is illustrated through synthetic examples in one and two dimensions. arXiv:1807.00393v2 [math.OC]From the samples provided, we seek a map T that would perform the transport well when applied to other independent realizations of the unknown distributions µ, ν. 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.

Distributional barycenter problem through data-driven flows

“…The methodology for the solution of the data-driven distributional barycenter problem proposed in this article can be used with general cost functions. It improves significantly over previous approaches to the barycenter problem based on adversarial games ( [7,28,34]). The latter have two players: one that proposes cost-minimizing maps through time-evolving flows, and another that builds test functions to enforce the push-forward condition.…”

Adaptation Capacity Management of the Passenger Air Carrier as a Participant in the Tourism Product Creation Process

“…The adaptive methodology for the optimal solution of the transport problem based on the sample by the standard quadratic cost function proposed by Essid et al [18] has significantly reduced data preparation and simplified its practical application. The adaptive approach of strategic planning of the airport can be considered as a component of development of its adaptation potential [19].…”

Conditional density estimation and simulation through optimal transport

“…Yet one may wish for a more adaptive approach, that will extract the relevant features from the data without any a priori knowledge of which could be relevant. One possibility is to extend to the barycenter problem the adaptive methodology recently developed for optimal transport in Essid et al (2018). Another is to replace the features G s (y) by low-rank factorizations, as is already done for the z-dependence of ψ in the current implementation.…”