Statistical Physics Approach to the Optimal Transport Problem

Originally defined for the optimal allocation of resources, optimal transport (OT) has found many theoretical and practical applications in multiple domains of science and physics. In this Letter we develop a new method for solving the discrete version of this problem using techniques derived from statistical physics. We derive a strongly concave free energy function that captures the constraints of the OT problem at a finite temperature. Its maximum defines an optimal transport plan, or registration between the two discrete probability measures that are compared, as well as a pseudodistance between those measures that satisfies the triangular inequalities. The computation of this pseudodistance is fast and numerically stable. The temperature dependent OT pseudodistance is shown to decrease monotonically with respect to the inverse of the temperature and to converge to the standard OT distance at zero temperature, providing a robust framework for temperature annealing. We illustrate applications of this framework to the problem of image comparison.