The infimal convolution structure of the Hellinger-Kantorovich distance

TL;DR

The paper rigorously proves that the Hellinger-Kantorovich distance $\mathsf{HK}$ between finite measures is the metric infimal convolution $\mathsf{He}_2 \nabla \mathsf{W}_2$, confirming a conjecture by LMSV. The approach embeds the problem in Unbalanced Optimal Transport through Marginal Entropy-Transport, recasting a single-step WHe-minimization as a cone-OT problem and exploiting a dynamic formulation of HK on the geometric cone. The authors establish two complementary inequalities—$\mathsf{He}_2 \nabla \mathsf{W}_2\le \mathsf{HK}$ and $\mathsf{HK}\le \mathsf{He}_2 \nabla \mathsf{W}_2$—by careful discretization, cone-based action estimates, and delicate measure-theoretic stability arguments, thereby deriving the equality. They also explore the general properties of infimal convolution of distances, its potential degeneracy, and connections to gradient-flow frameworks, paving the way for extensions to other exponents $p$ and to broader metric settings.

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.

Figures (1)

• Figure 1: A representation of the space $Y$.

Theorems & Definitions (68)

• Theorem 1.1
• Remark 2.1: Minimal radius of non-degenerate geodesics
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Example 3.4: Inf-convolution of distances may take the value $+\infty$
• Proposition 3.5
• proof