---
title: Spherical Hellinger–Kantorovich Distance
url: https://www.emergentmind.com/topics/spherical-hellinger-kantorovich-distance
type: topic
---

# Spherical Hellinger–Kantorovich Distance

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 $\mathcal P(X)$, restricted to the unit sphere $\mathcal S(C(X))$ [2303.13901, 1508.07941, 1712.01888].

The restriction to $\mathcal P(X)$ 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” [2303.13901, 2207.09815].

## 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
   $$
   C(X) = (X \times [0,\infty)) / (X \times \{0\} \to o)
   $$
   with cone distance
   $$
   d_{C}^2([x_0,r_0],[x_1,r_1]) = r_0^2 + r_1^2 - 2r_0 r_1 \cos( \min\{ d(x_0,x_1), \pi \} ).
   $$
   The SHK squared distance between $\mu_0, \mu_1 \in \mathcal P(X)$ is
   $$
   \operatorname{SHK}^2(\mu_0, \mu_1) = \min \{ W_2^2(\lambda_0, \lambda_1): \lambda_i \in \mathcal P(C(X)),\ \pi^x_\#(r^2 \lambda_i) = \mu_i \},
   $$
   where $W_2^2$ is the usual 2-Wasserstein distance on the cone [1508.07941, 1509.00068, 1712.01888].

2. **Entropy–transport formulation:** On $X \times X$,
   $$
   \inf_{\eta \geq 0}
   \Bigg\{
     \int_X B \left( \frac{d\eta_0}{d\mu_0} \right) d\mu_0
   + \int_X B \left( \frac{d\eta_1}{d\mu_1} \right) d\mu_1
   + \int_{X\times X} c(x,y) d\eta(x,y)
   \Bigg\},
   $$
   with $B(s)=s\log s - s+1$, and cost $c(x,y) = -2 \log \cos( d(x,y) )$ for $d(x,y) < \pi/2$ ($+\infty$ otherwise) [1509.00068].

3. **Benamou–Brenier (dynamical) formulation:** For curves $(\mu_t)_{t\in[0,1]}$ in $\mathcal P(X)$,
   $$
   \operatorname{SHK}_{\kappa}^2(\mu_0,\mu_1)
   = \inf \int_0^1 \int_X
     \left[ \| v_t(x) \|^2 + \frac{\kappa^2}{4} \alpha_t(x)^2 \right] d\mu_t(x) dt,
   $$
   under the continuity equation
   $$
   \partial_t \mu_t + \mathrm{div}( v_t \mu_t ) = \alpha_t \mu_t, \qquad \int_X \alpha_t\, d\mu_t = 0,
   $$
   ensuring local mass change with global conservation [2303.13901, 2603.20656, 2207.09815, 1509.00068].

The table below summarizes these main equivalences:

| Formulation        | Domain      | Mass constraint      |
|--------------------|------------|---------------------|
| Cone–lift          | $C(X)$     | $\| \cdot \|_1 = 1$ |
| Entropy–transport  | $X \times X$ | $|\mu| = |\nu| = 1$ |
| Benamou–Brenier    | $X$        | $\int \alpha_t d\mu_t = 0$ |

## 3. Geodesic Structure and Riemannian Geometry

SHK endows $\mathcal P(X)$ with a natural geodesic space structure. Geodesics are obtained as projections of constant-speed geodesics in the cone $(\mathcal M^+, {\rm HK})$, reparametrized and renormalized to fix total mass at all times. Explicitly, if $(\mu_t, v_t, \alpha_t)$ solves the HK dynamical problem, then the corresponding SHK geodesic $(\mu_t^S, v_t^S, \alpha_t^S)$ is constructed via time-rescaling and normalization:
$$
\mu_t^S = \frac{ \mu_{s(t)} }{ M(s(t)) }, \qquad M(s) = 1 - s(1-s) \operatorname{HK}_\kappa^2/ \kappa^2,
$$
with $s(t)$ a sinusoidal reparametrization and corresponding transformations for $v_t^S$ and $\alpha_t^S$ [2303.13901].

The tangent space at $\mu \in \mathcal P(X)$ is
$$
T_\mu \mathcal P(X) = \{ (v,\alpha) \in L^2(TX;\mu) \times L^2(\mu)\ :\ \int \alpha\, d\mu = 0 \},
$$
endowed with the inner product
$$
\langle (v,\alpha), (w,\beta) \rangle_{T_\mu}
= \int_X \langle v,w \rangle\, d\mu + \frac{\kappa^2}{4} \int_X \alpha\, \beta\, d\mu.
$$
This structure admits well-defined exponential and logarithmic maps, consistent discretizations, and a left-inverse identity [2303.13901].

## 4. Special Cases, Limits, and Geometric Properties

- **Dirac masses:** For Dirac measures $\delta_x, \delta_y \in \mathcal P(X)$, $\operatorname{SHK}_\kappa( \delta_x, \delta_y ) = d(x,y )$, recovering the 2-Wasserstein geometry for atomic probabilities [2303.13901, 2207.09815, 1509.00068].
- **Convexity and curvature:** $(\mathcal P(X), \operatorname{SHK})$ is a positively curved metric space in the sense of Alexandrov if $(X,d)$ has curvature $\ge 1$ [1508.07941, 1712.01888]. The squared distance is jointly convex and semiconcave on sets of measures with densities uniformly bounded above and below.
- **Comparison with Wasserstein and Hellinger:** $\operatorname{SHK}$ lies between Hellinger–Kakutani and $W_2$ distances. As the geometry is “blown up” ($d \to \infty$), SHK converges to the Hellinger–Kakutani; “shrinking” distances ($d \to d/\lambda$) yields $W_2$ in the limit [1508.07941].
- **Forbidden regions:** For $d(x,y) > \pi/2$, pure transportation is penalized with infinite cost; mass must be split, created, or annihilated [1509.00068, 2207.09815].

## 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 $\mathcal E$ (with first variation $f$) is
$$
\mathrm{grad}_{\operatorname{SHK}} \mathcal{E}(u) = -\nabla \cdot (u \nabla f) + u (f - \bar f),
$$
with $\bar f = \int u f$. This leads to evolution equations of the form
$$
\partial_t u = -\operatorname{grad}_{\operatorname{SHK}} \mathcal E(u) = -\nabla \cdot(u \nabla f) + u (f - \bar f)
$$
[1809.03430, 2603.20656].

Key functional inequalities include a generalized log-Sobolev (entropy–entropy-production) inequality
$$
\int E(x,u) dx \le C \left[ \int u |\nabla f|^2 dx + \int u (f-\bar f)^2 dx \right],
$$
and Talagrand-type transportation inequalities:
$$
\operatorname{SHK}^2(u, m) \le C \mathcal E[u].
$$
These are crucial for establishing convergence rates and well-posedness of gradient flows in SHK geometry [1809.03430, 2207.09815].

The “minimizing movement” (JKO) scheme applied to $\operatorname{SHK}$ produces discrete-time approximations to gradient flows, with each time step defined as a proximal minimization
$$
\nu_n \in \operatorname{Argmin}_{\nu \in \mathcal P(X)} \left\{ \frac{1}{2\tau} \operatorname{SHK}^2(\nu, \nu_{n-1}) + \mathcal E(\nu) \right\},
$$
and uniform density bounds and geometric convexity conditions ensuring convergence to Evolutionary Variational Inequality (EVI) solutions [2207.09815].

## 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 [2603.20656]. 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 [2303.13901].

On the technical side, discrete barycentric projections and log/exponential maps are developed for practical computation, with convergence guarantees [2303.13901].

## 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 $\mathcal P(X)$. 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:
$$
\operatorname{HK}_\kappa^2(r_0 \sigma_0, r_1 \sigma_1)
= r_0^2 + r_1^2 - 2 r_0 r_1 \cos \left( \frac{ \operatorname{SHK}_\kappa(\sigma_0,\sigma_1) }{ \kappa } \right)
$$
[2303.13901, 1508.07941, 1509.00068, 1712.01888]. The SHK thus interpolates between unbalanced transport (HK), classical balanced transport ($W_2$), 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.

Source: https://www.emergentmind.com/topics/spherical-hellinger-kantorovich-distance