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Quadratic optimal transportation problem with a positive semi definite structure on the cost function

Seonghyeon Jeong

Abstract

Optimal transportation problem seeks for a coupling $π$ of two probability measures $μ$ and $ν$ which minimize the total cost $\int c dπ$, which is linear in $π$. In this paper, we introduce a variation of optimal transportation problem which we call quadratic transportation problem that considers a total cost $\iint c dπdπ$ which is quadratic in $π$. We compare this problem with other variations of optimal transportation problem, and prove some properties of the solutions to the problem. Then, we introduce squared cost function, which let us consider the total cost $\iint c dπdπ$ as a positive semi-definite bilinear operator on probability measures, and show Kantorovich duality formula when we have a squared cost function.

Quadratic optimal transportation problem with a positive semi definite structure on the cost function

Abstract

Optimal transportation problem seeks for a coupling of two probability measures and which minimize the total cost , which is linear in . In this paper, we introduce a variation of optimal transportation problem which we call quadratic transportation problem that considers a total cost which is quadratic in . We compare this problem with other variations of optimal transportation problem, and prove some properties of the solutions to the problem. Then, we introduce squared cost function, which let us consider the total cost as a positive semi-definite bilinear operator on probability measures, and show Kantorovich duality formula when we have a squared cost function.
Paper Structure (17 sections, 22 theorems, 139 equations)

This paper contains 17 sections, 22 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Notations
  4. Quadratic transportation problem
  5. Quadratic transportation problem
  6. Comparison with other variations of OT
  7. Classical optimal transportation problems
  8. Multi-marginal problems
  9. Co-optimal transportation problems
  10. Other transportation problems
  11. Elementary properties
  12. Existence, non-uniqueness, and non-locality
  13. Monotonicity and positive semi-definite costs
  14. Duality for the squared costs
  15. Classical optimal transportation problem
  16. ...and 2 more sections

Key Result

Theorem 1

For squared cost function $c$, we have the following identity where the infimum and supremum are taken over some admissible sets.

Theorems & Definitions (52)

  • Theorem : Duality, rough statement
  • Example 1
  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Example 2
  • Example 3
  • Lemma 3.1
  • proof
  • ...and 42 more