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Physics approach to the variable-mass optimal-transport problem.

Patrice Koehl1, Marc Delarue2, Henri Orland3

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Summary
This summary is machine-generated.

This study introduces a variable-mass optimal transport (VMOT) method using statistical physics, enabling comparisons of unbalanced distributions. This approach offers a competitive and robust framework for diverse data science applications.

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Area of Science:

  • Probability and Statistics
  • Machine Learning
  • Data Science

Background:

  • Optimal transport (OT) traditionally requires distributions with equal mass.
  • Real-world applications often involve unbalanced distributions, necessitating mass creation/destruction.
  • Existing OT methods are limited by the balance condition, restricting partial matching applications.

Purpose of the Study:

  • To propose and formalize a generalized optimal transport approach for unbalanced distributions (variable-mass optimal-transport, VMOT).
  • To adapt statistical physics techniques for solving the VMOT problem.
  • To develop a robust framework for partial matching problems.

Main Methods:

  • Developed a discrete variable-mass optimal-transport (VMOT) formalism using statistical physics.
  • Derived a strongly concave effective free-energy function for VMOT constraints at finite temperatures.
  • Introduced a temperature-dependent optimal transport distance that monotonically decreases to the standard VMOT distance.

Main Results:

  • The proposed method provides a weak distance (divergence) between unbalanced distributions.
  • The temperature-dependent distance offers a robust framework for temperature annealing.
  • The VMOT implementation exhibits competitive time complexity similar to regularized OT algorithms.

Conclusions:

  • The statistical physics-based VMOT framework effectively addresses the limitations of traditional optimal transport for unbalanced distributions.
  • This approach is computationally competitive and applicable to problems like partial shape matching.
  • The method provides a generalized and robust solution for comparing distributions with varying masses.