Hellinger–Kantorovich Metric in Optimal Transport

• The Hellinger–Kantorovich metric is defined as the infimal convolution of Wasserstein and Hellinger distances, capturing both mass transport and reaction processes.
• Its dynamic formulation via a Benamou–Brenier approach rigorously describes geodesic paths in spaces of nonnegative measures with non-conserved mass.
• The integration of marginal entropy–transport and cone formulations offers practical insights for tackling unbalanced optimal transport problems in various applications.

The Hellinger–Kantorovich (HK) metric is a geodesic distance on the space of nonnegative Radon measures that unifies optimal transport (Wasserstein) and unbalanced information-geometric divergences (notably, the Hellinger and Fisher–Rao distances). It metrizes the weak topology, admits rich dual, dynamic, and geometric formulations, and provides a rigorous framework for comparing and evolving measures when total mass is not conserved. The HK metric is fundamentally characterized as the infimal convolution—in a metric sense—of the quadratic Wasserstein and Hellinger distances, and is a cornerstone of modern unbalanced optimal transport theory.

1. Infimal Convolution Structure

The HK metric can be concretely expressed as a metric infimal convolution of the Hellinger and Wasserstein distances (Ponti et al., 17 Mar 2025). For $\mu_0, \mu_1$ nonnegative Borel measures, define a multi-step chain of intermediate measures,

$\mu_0, \nu_1, \mu_1, \nu_2, \ldots, \nu_N, \mu_N$

with $\mu_0 = \mu_0$, %%%%2%%%%, and consider the energy functional: $$\mathcal{E}_N = N\sum_{i=1}^N \left[ \mathrm{He}^2(\mu_{i-1}, \nu_i) + \mathrm{W}^2(\nu_i, \mu_i) \right]$$ where $\mathrm{He}$ denotes the Hellinger distance and $\mathrm{W}$ the Wasserstein distance. Then, taking the limit as $N \to \infty$,

$$(\mathrm{He} \nabla \mathrm{W})^2(\mu_0, \mu_1) = \liminf_{N \to \infty} \inf \mathcal{E}_N$$

one recovers exactly the HK distance $HK^2(\mu_0, \mu_1)$. At each infinitesimal step, the path evolves first by a "reaction" (mass adjustment, Hellinger-type), followed by a "transport" (mass movement, Wasserstein-type), and in the limit these steps synthesize the HK geometry.

2. Marginal Entropy–Transport Problem

A crucial ingredient is the so-called marginal entropy–transport problem, which serves as the building block for the infimal convolution (Ponti et al., 17 Mar 2025). Given $\mu_0, \mu_1 \in \mathcal{M}_+(X)$,

$$\mathrm{WHe}(\mu_0, \mu_1) = \inf_{\nu \in \mathcal{M}_+(X)} \left\{ \mathrm{He}^2(\mu_0, \nu) + \mathrm{W}^2(\nu, \mu_1) \right\}$$

represents the minimal cost of "reacting" from $\mu_0$ to an intermediate measure $\nu$ (via Hellinger) and then "transporting" $\nu$ to $\mu_1$ (via Wasserstein). The properties, existence, and explicit form of optimal $\nu$ are analyzed: for atomic measures, the minimizer typically splits mass depending on the Hellinger penalty and underlying transport cost, offering crucial insight into the composition of the HK metric.

3. Unbalanced Optimal Transport: Cone and Dynamic Formulations

The interpretation of HK as a unifying structure between reaction and transport leverages the geometric cone formalism and dynamical approaches from unbalanced optimal transport (UOT) (Liero et al., 2015, Liero et al., 2015). The cone over $X$,

$$\mathfrak{C}[X] = (X \times [0, \infty)) \big/ (X \times \{0\})$$

serves as the ambient space for "lifting" measures, with the cone distance

$$\mathsf{H}([x_0, r_0], [x_1, r_1]) = r_0^2 + r_1^2 - 2 r_0 r_1 \cos(\min\{|x_0 - x_1|, \pi/2\})$$

which restricts transport to within a "cone angle" ($\pi/2$ threshold).

The dynamic HK distance admits a Benamou–Brenier-type formula (Liero et al., 2015): $$HK^2(\mu_0, \mu_1) = \inf\left\{ \int_0^1 \int_X \left( |v_t|^2 + \frac{1}{4} |w_t|^2 \right) d\mu_t dt : \partial_t \mu_t + \nabla\cdot(v_t\mu_t) = w_t\mu_t \right\}$$ where $v_t$ is a velocity field (transport) and $w_t$ a growth/shrinkage field (reaction). This formulation makes explicit the combined geometric (Wasserstein) and information-theoretic (Hellinger) effects embedded within HK.

4. Estimates, Convergence, and Infimal Convolution Limit

Establishing the equivalence of the infimal convolution and dynamic characterizations of the HK metric requires precise estimates (Ponti et al., 17 Mar 2025). The paper constructs approximations, discretizing optimal dynamic curves into $N$-step chains and demonstrating that the limit of the discrete energy equals the geodesic action for HK: $$\liminf_{N\to\infty} \mathcal{E}_N \le HK^2(\mu_0, \mu_1)$$ via uniform control of intermediate masses and a careful partition of "good" and "bad" segments (the latter compensated via measurable geodesic selections and dilation—key for the compactness argument). The converse inequality is also shown, using minimization over chains to produce continuous curves in the cone whose energy bounds the dynamic cost from below, yielding a tight characterization.

5. Abstract Metric Infimal Convolution: Generalization

Extending beyond the HK context, the analysis provides an abstract infimal convolution construction for general pairs of (possibly non-metric) cost functions $\phi_1, \phi_2$ on an arbitrary space $U$ (Ponti et al., 17 Mar 2025): $$(\phi_1 \nabla \phi_2)(z_0, z_1) = \liminf_{N\to\infty} \inf\left\{ N\sum_{i=1}^N (\phi_1^2(x_{i-1}, y_i) + \phi_2^2(y_i, x_i)) : \text{chains from } z_0 \text{ to } z_1\right\}$$ This construction recovers the standard (convex, Hilbertian) infimal convolution when $\phi_1$ and $\phi_2$ are induced by norms, and, under mild conditions, remains a metric (non-negativity, triangle inequality) in the limit. Such a perspective suggests broad applicability of the concatenation-of-processes (reaction/transport) idea for constructing new metrics in other geometric and analytic settings.

6. Significance: Geometry, Applications, and Future Directions

Recognizing the HK metric as the metric infimal convolution of Hellinger and Wasserstein distances has profound consequences. It clarifies the mathematical structure of geodesics, cost-splitting, and the behavior of interpolations between measures in the presence of both transport and mass variation. This dual structure is pivotal in

• gradient flows on measure spaces (reaction-diffusion PDEs, unbalanced Fokker–Planck, dissipative systems),
• barycenter and clustering problems, where the nuanced local-to-global transition between clustering and transport is driven by the critical length scale,
• universal contraction and regularity estimates (notably in metric measure spaces with curvature lower bounds) (Luise et al., 2019),
• the rigorous comparison and embedding of statistical divergences and transport costs (Ponti, 2019).

The general theory may be further developed for matrix-valued measures (Kantorovich–Bures metric (Brenier et al., 2018)), for potentially non-symmetric or non-conservative cost structures, and in more abstract measure or function spaces. The infimal convolution structure is a robust blueprint for unifying reaction, absorption, creation, and geometric transport effects in metric measure theory.

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Summary Table: Core Formulations for HK as Infimal Convolution

Structure	Formula	Section Reference
Discrete chain energy	$$N \sum_{i=1}^N \left[\mathrm{He}^2(\mu_{i-1}, \nu_i) + \mathrm{W}^2(\nu_i, \mu_i)\right]$$	Infimal Convolution, 1
1-step marginal entropy-transport	$$\inf_{\nu} \left[\mathrm{He}^2(\mu_0, \nu) + \mathrm{W}^2(\nu, \mu_1)\right]$$	Marginal ET Problem, 2
Abstract ICC	$$\liminf_{N\to\infty} \inf\; N\sum [ \phi_1^2(x_{i-1}, y_i) + \phi_2^2(y_i, x_i) ]$$	Abstract ICC, 5
Dynamic (BB) Formulation	$$\min \int_0^1 \int (|v_t|^2 + \tfrac14 |w_t|^2)\; d\mu_t dt$$	Cone/Dynamic, 3
Cone cost	$r_0^2 + r_1^2 - 2r_0 r_1 \cos(\cdot \wedge \pi/2)$	Cone/Dynamic, 3

This structure identifies the HK metric as the precise metric-theoretic fusion of mass transport and reaction, making it a central object in modern analysis and geometry on spaces of measures.