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arXiv · 2309.04846

Non-commutative Optimal Transport for semi-definite positive matrices

Augusto Gerolin·Nataliia Monina
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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

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Augusto Gerolin, Nataliia Monina. 2023-09-09. Non-commutative Optimal Transport for semi-definite positive matrices. https://arxiv.org/abs/2309.04846

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