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Non-commutative Optimal Transport for semi-definite positive matrices

Augusto Gerolin, Nataliia Monina

Abstract

We introduce the von Neumann entropy regularization of Unbalanced Non-commutative Optimal Transport, specifically Non-commutative Optimal Transport between semi-definite positive matrices (not necessarily with trace one). We prove the existence of a minimizer, compute the weak dual formulation and prove $Γ$-convergence results, demonstrating convergence to both Unbalanced Non-commutative Optimal Transport (as the Entropy-regularization parameter tends to zero) and von Neumann entropy regularized Non-commutative Optimal Transport problems (as the unbalanced penalty parameter tends to infinity). To draw an analogy to the Non-commutative case, we provide a concise introduction of the static formulation of Unbalanced Optimal Transport between positive measures and bounded cost

Non-commutative Optimal Transport for semi-definite positive matrices

Abstract

We introduce the von Neumann entropy regularization of Unbalanced Non-commutative Optimal Transport, specifically Non-commutative Optimal Transport between semi-definite positive matrices (not necessarily with trace one). We prove the existence of a minimizer, compute the weak dual formulation and prove -convergence results, demonstrating convergence to both Unbalanced Non-commutative Optimal Transport (as the Entropy-regularization parameter tends to zero) and von Neumann entropy regularized Non-commutative Optimal Transport problems (as the unbalanced penalty parameter tends to infinity). To draw an analogy to the Non-commutative case, we provide a concise introduction of the static formulation of Unbalanced Optimal Transport between positive measures and bounded cost
Paper Structure (8 sections, 15 theorems, 102 equations)

This paper contains 8 sections, 15 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Unbalanced Optimal Transport
  3. Shannon-Entropy regularized Unbalanced Optimal Transport
  4. Dual problem
  5. Unbalanced Non-commutative Optimal Transport
  6. Weak duality
  7. Existence of a minimizer
  8. Gamma-convergence when $\mathop{\mathrm{\varepsilon}}\nolimits\to 0^+$ and when $\tau_1=\tau_2\to +\infty$

Key Result

Proposition 2.1

Let $\mathop{\mathrm{\varepsilon}}\nolimits,\tau_1, \tau_2>0$ be the positive parameters, $X$ and $Y$ be complete separable metric spaces, $\mu \in \mathcal{M}_+( X ),~ \nu \in \mathcal{M}_+( Y )$ be positive measures, and let $c\in L^\infty(X\times Y)$ be a cost function. If $u \in L^\infty( X ),~

Theorems & Definitions (36)

  • Definition 2.1: Unbalanced Entropic $c$-transform or $(c,\tau,\mathop{\mathrm{\varepsilon}}\nolimits)$-transform
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 26 more