Non-commutative Optimal Transport for semi-definite positive matrices

TL;DR

This paper develops a von Neumann entropy regularization framework for unbalanced non-commutative optimal transport between semi-definite positive matrices, establishing theoretical properties and convergence results.

Contribution

It introduces a novel entropy regularization for non-commutative optimal transport and proves key theoretical properties including existence, duality, and convergence.

Findings

Proved existence of minimizers for the regularized problem

Derived the weak dual formulation and $b3$-convergence results

Established convergence to unbalanced non-commutative optimal transport

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 $\Gamma$-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