Calculus of Variations and Geometric Measure Theory

A. Gerolin - N. Monina

Non-commutative Optimal Transport for semi-definite positive matrices

created by gerolin on 28 Feb 2024



Inserted: 28 feb 2024

Year: 2023

ArXiv: 2309.04846 PDF


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