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Statistical Physics Approach to the Optimal Transport Problem.

Patrice Koehl1, Marc Delarue2, Henri Orland3

  • 1Department of Computer Science and Genome Center, University of California, Davis, California 95616, USA.

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|September 7, 2019
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Summary
This summary is machine-generated.

This study introduces a novel statistical physics method for optimal transport (OT), enabling fast and stable computation of a pseudodistance between probability measures. This approach offers a robust framework for image comparison and other applications.

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Area of Science:

  • Statistical physics
  • Computational mathematics
  • Image analysis

Background:

  • Optimal transport (OT) is crucial for resource allocation and has broad applications in science and physics.
  • Existing methods for discrete OT can be computationally intensive and lack numerical stability.
  • There is a need for efficient and robust algorithms to solve discrete OT problems.

Purpose of the Study:

  • To develop a new statistical physics-based method for solving the discrete optimal transport problem.
  • To introduce a temperature-dependent pseudodistance metric for comparing probability measures.
  • To demonstrate the utility of this framework in image comparison tasks.

Main Methods:

  • Derivation of a strongly concave free energy function capturing OT constraints at finite temperatures.
  • Utilizing the maximum of this free energy to define an optimal transport plan and a pseudodistance.
  • Employing techniques from statistical physics to ensure computational speed and numerical stability.

Main Results:

  • A novel, fast, and numerically stable method for discrete optimal transport.
  • A temperature-dependent pseudodistance that satisfies triangular inequalities.
  • Demonstration that the pseudodistance monotonically decreases with inverse temperature, converging to the standard OT distance.

Conclusions:

  • The developed method provides a robust framework for optimal transport, particularly suitable for temperature annealing.
  • The new pseudodistance metric offers an efficient and stable alternative for comparing probability measures.
  • The approach shows promise for applications such as image comparison and registration.