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arXiv · 2503.12939

The infimal convolution structure of the Hellinger-Kantorovich distance

Nicolò De Ponti·Giacomo Enrico Sodini·Luca Tamanini
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Abstract

We show that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances, as conjectured by Liero, Mielke, and Savaré. To prove it, we study with the tools of Unbalanced Optimal Transport the so called Marginal Entropy-Transport problem that arises as a single minimization step in the definition of infimal convolution. Careful estimates and results when the number of minimization steps diverges are also provided, both in the specific case of the Hellinger-Kantorovich setting and in the general one of abstract distances.

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Nicolò De Ponti, Giacomo Enrico Sodini, Luca Tamanini. 2025-03-17. The infimal convolution structure of the Hellinger-Kantorovich distance. https://arxiv.org/abs/2503.12939

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