Bayesian Inference for Optimal Transport with Stochastic Cost

TL;DR

This paper introduces a Bayesian framework for optimal transport with stochastic costs, enabling inference of transport plan distributions and incorporating prior knowledge, with a tailored HMC sampling method.

Contribution

It presents a novel Bayesian approach to stochastic optimal transport, modeling the transport plan distribution and developing a specialized HMC sampling technique.

Findings

Effective inference of transport plan distributions under stochastic costs

Incorporation of prior information into optimal transport models

Development of a tailored HMC sampling method for the posterior

Abstract

In machine learning and computer vision, optimal transport has had significant success in learning generative models and defining metric distances between structured and stochastic data objects, that can be cast as probability measures. The key element of optimal transport is the so called lifting of an \emph{exact} cost (distance) function, defined on the sample space, to a cost (distance) between probability measures over the sample space. However, in many real life applications the cost is \emph{stochastic}: e.g., the unpredictable traffic flow affects the cost of transportation between a factory and an outlet. To take this stochasticity into account, we introduce a Bayesian framework for inferring the optimal transport plan distribution induced by the stochastic cost, allowing for a principled way to include prior information and to model the induced stochasticity on the transport plans. Additionally, we tailor an HMC method to sample from the resulting transport plan posterior distribution.