Spherical Hellinger–Kantorovich Distance

• Spherical Hellinger–Kantorovich distance is a geodesic metric on probability measures integrating optimal transport with creation and depletion of mass while conserving total mass.
• It unifies 2-Wasserstein and Hellinger/Fisher–Rao geometries through equivalent static (cone-lift, entropy–transport) and dynamical (Benamou–Brenier) formulations.
• Its robust geometric structure supports gradient flows and computational implementations, enabling scalable analysis in data science and nonlinear PDE modeling.

The spherical Hellinger–Kantorovich distance is a geodesic metric on the space of probability measures that integrates optimal transport with creation and depletion of mass, subject to conservation of total mass. It arises as the restriction of the unbalanced Hellinger–Kantorovich metric to the probability simplex, equipping the latter with a natural “spherical” geometry that captures both transport and reaction mechanisms. This construction unifies and interpolates between 2-Wasserstein (optimal transport) and Hellinger/Fisher–Rao geometries, yielding a distance that is especially well-behaved for probability measures on compact metric spaces, such as spheres, convex domains, or manifolds.

1. Conceptual Foundations and Definition

The spherical Hellinger–Kantorovich (SHK) distance, denoted $\operatorname{SHK}_\kappa$, is defined on the set $\mathcal P(X)$ of Borel probability measures over a compact metric space $(X,d)$. It is canonically derived from the Hellinger–Kantorovich (HK) distance on finite nonnegative measures $\mathcal M^+(X)$ by restricting consideration to probability measures and enforcing a total-mass constraint along geodesics.

Given $\kappa > 0$, the static “soft-marginal” primal form of the HK distance is

$$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$

with $$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$ for $d(x,y) < \kappa \pi/2$, and $+\infty$ otherwise. On probability measures, the spherical Hellinger–Kantorovich distance is then defined as

$$\operatorname{SHK}_\kappa(\mu_0, \mu_1) = \kappa \arccos \left[ 1 - \frac{ \operatorname{HK}_\kappa^2(\mu_0,\mu_1) }{2\kappa^2} \right],$$

which measures the “angle” between elements in the positive-mass cone over $\mathcal P(X)$0, restricted to the unit sphere $\mathcal P(X)$1 (Sarrazin et al., 2023, Liero et al., 2015, Laschos et al., 2017).

The restriction to $\mathcal P(X)$2 imposes conservation of total mass, in contrast to the unbalanced HK metric, under which geodesics can leave the simplex and return as mass is temporarily “created” or “annihilated” (Sarrazin et al., 2023, Laschos et al., 2022).

2. Equivalent Characterizations

The SHK distance admits several equivalent static and dynamic formulations:

1. Cone–lift (static) formulation: Probabilities are lifted to measures on the unit sphere of the metric cone

$\mathcal P(X)$3

with cone distance

$\mathcal P(X)$4

The SHK squared distance between $\mathcal P(X)$5 is

$\mathcal P(X)$6

where $\mathcal P(X)$7 is the usual 2-Wasserstein distance on the cone (Liero et al., 2015, Liero et al., 2015, Laschos et al., 2017).

2. Entropy–transport formulation: On $\mathcal P(X)$8,

$\mathcal P(X)$9

with $(X,d)$0, and cost $(X,d)$1 for $(X,d)$2 ($(X,d)$3 otherwise) (Liero et al., 2015).

3. Benamou–Brenier (dynamical) formulation: For curves $(X,d)$4 in $(X,d)$5,

$(X,d)$6

under the continuity equation

$(X,d)$7

ensuring local mass change with global conservation (Sarrazin et al., 2023, Mustafi et al., 21 Mar 2026, Laschos et al., 2022, Liero et al., 2015).

The table below summarizes these main equivalences:

Formulation	Domain	Mass constraint
Cone–lift	$(X,d)$8	$(X,d)$9
Entropy–transport	$\mathcal M^+(X)$0	$\mathcal M^+(X)$1
Benamou–Brenier	$\mathcal M^+(X)$2	$\mathcal M^+(X)$3

3. Geodesic Structure and Riemannian Geometry

SHK endows $\mathcal M^+(X)$4 with a natural geodesic space structure. Geodesics are obtained as projections of constant-speed geodesics in the cone $\mathcal M^+(X)$5, reparametrized and renormalized to fix total mass at all times. Explicitly, if $\mathcal M^+(X)$6 solves the HK dynamical problem, then the corresponding SHK geodesic $\mathcal M^+(X)$7 is constructed via time-rescaling and normalization:

$\mathcal M^+(X)$8

with $\mathcal M^+(X)$9 a sinusoidal reparametrization and corresponding transformations for $\kappa > 0$0 and $\kappa > 0$1 (Sarrazin et al., 2023).

The tangent space at $\kappa > 0$2 is

$\kappa > 0$3

endowed with the inner product

$\kappa > 0$4

This structure admits well-defined exponential and logarithmic maps, consistent discretizations, and a left-inverse identity (Sarrazin et al., 2023).

4. Special Cases, Limits, and Geometric Properties

• Dirac masses: For Dirac measures $\kappa > 0$5, $\kappa > 0$6, recovering the 2-Wasserstein geometry for atomic probabilities (Sarrazin et al., 2023, Laschos et al., 2022, Liero et al., 2015).
• Convexity and curvature: $\kappa > 0$7 is a positively curved metric space in the sense of Alexandrov if $\kappa > 0$8 has curvature $\kappa > 0$9 (Liero et al., 2015, Laschos et al., 2017). The squared distance is jointly convex and semiconcave on sets of measures with densities uniformly bounded above and below.
• Comparison with Wasserstein and Hellinger: $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$0 lies between Hellinger–Kakutani and $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$1 distances. As the geometry is “blown up” ($$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$2), SHK converges to the Hellinger–Kakutani; “shrinking” distances ($$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$3) yields $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$4 in the limit (Liero et al., 2015).
• Forbidden regions: For $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$5, pure transportation is penalized with infinite cost; mass must be split, created, or annihilated (Liero et al., 2015, Laschos et al., 2022).

5. Gradient Flows, Functional Inequalities, and Minimizing Movements

Gradient flows in SHK geometry correspond to solutions of nonlinear PDEs, often interpreted as degenerate Fokker–Planck equations. The SHK gradient of an entropy $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$6 (with first variation $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$7) is

$$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$8

with $$\operatorname{HK}_{\kappa}^2(\mu_0,\mu_1) = \inf_{\pi \in \mathcal M_+(X \times X)} \Bigl[ \int c_\kappa^{\mathrm{HK}}(x,y)\, d\pi(x,y) + \kappa^2 \operatorname{KL}(\pi_0|\mu_0) + \kappa^2 \operatorname{KL}(\pi_1|\mu_1) \Bigr],$$9. This leads to evolution equations of the form

$$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$0

(Kondratyev et al., 2018, Mustafi et al., 21 Mar 2026).

Key functional inequalities include a generalized log-Sobolev (entropy–entropy-production) inequality

$$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$1

and Talagrand-type transportation inequalities:

$$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$2

These are crucial for establishing convergence rates and well-posedness of gradient flows in SHK geometry (Kondratyev et al., 2018, Laschos et al., 2022).

The “minimizing movement” (JKO) scheme applied to $$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$3 produces discrete-time approximations to gradient flows, with each time step defined as a proximal minimization

$$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$4

and uniform density bounds and geometric convexity conditions ensuring convergence to Evolutionary Variational Inequality (EVI) solutions (Laschos et al., 2022).

6. Applications and Computational Aspects

Recent work exploits the SHK metric structure for associative memory models based on Sinkhorn divergences, with dynamics realized as gradient flows in SHK geometry. These algorithms update both support locations and weights of point clouds and show robust geometric convergence under suitable conditions (Mustafi et al., 21 Mar 2026). In data analysis contexts, the SHK distance, by linearizing the tangent space structure, allows for scalable implementations and for integration into methods requiring a linear metric geometry (Sarrazin et al., 2023).

On the technical side, discrete barycentric projections and log/exponential maps are developed for practical computation, with convergence guarantees (Sarrazin et al., 2023).

7. Relationship to Optimal Transport, Hellinger, and Cone Geometry

The SHK metric provides a canonical realization of the “spherical part” of the generalized Hellinger–Kantorovich cone over $$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$5. The SHK metric can be viewed as the angle in the metric cone (for mass-varying HK), with the law of cosines explicitly relating the distances:

$$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$6

(Sarrazin et al., 2023, Liero et al., 2015, Liero et al., 2015, Laschos et al., 2017). The SHK thus interpolates between unbalanced transport (HK), classical balanced transport ($$c_\kappa^{\mathrm{HK}}(x,y) = -2\kappa^2 \log \cos ( d(x,y)/\kappa )$$7), and multiplicative-mass transport (Hellinger), and precisely quantifies the “angle” between probability distributions on a metric space.

Throughout, the SHK distance retains the geometric structure required for robust variational analysis, gradient flows, and dynamic interpolation, making it an indispensable tool in modern unbalanced transport theory.