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Sinkhorn algorithms and linear programming solvers for optimal partial transport problems

Yikun Bai

Abstract

In this note, we generalize the classical optimal partial transport (OPT) problem by modifying the mass destruction/creation term to function-based terms, introducing what we term ``generalized optimal partial transport'' problems. We then discuss the dual formulation of these problems and the associated Sinkhorn solver. Finally, we explore how these new OPT problems relate to classical optimal transport (OT) problems and introduce a linear programming solver tailored for these generalized scenarios.

Sinkhorn algorithms and linear programming solvers for optimal partial transport problems

Abstract

In this note, we generalize the classical optimal partial transport (OPT) problem by modifying the mass destruction/creation term to function-based terms, introducing what we term ``generalized optimal partial transport'' problems. We then discuss the dual formulation of these problems and the associated Sinkhorn solver. Finally, we explore how these new OPT problems relate to classical optimal transport (OT) problems and introduce a linear programming solver tailored for these generalized scenarios.
Paper Structure (9 sections, 4 theorems, 38 equations)

This paper contains 9 sections, 4 theorems, 38 equations.

Table of Contents

  1. Notation and assumptions
  2. Outlines
  3. Background
  4. Entropy Function and $f$-Divergence
  5. Optimal Transport and Optimal Partial Transport
  6. A Generalized Version of the OPT Problem
  7. Entropic Regularization
  8. Dual formulation of entropic (generalized) OPT problems
  9. Background: convex function, conjugate, dual formulation.

Key Result

Lemma 1

We claim eq:TV* holds.

Theorems & Definitions (17)

  • Example 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • ...and 7 more