Hellinger Discrepancy Tensor in HK Geometry

• Hellinger Discrepancy Tensor is the HK Riemannian metric at a reference measure that unifies mass transport and reaction effects.
• It supports logarithmic and exponential maps to linearize the HK distance, enabling local Hilbertian embeddings and efficient discrete data analysis.
• The tensor bridges Wasserstein and Fisher–Rao geometries via an intrinsic scale κ, providing exact decompositions of transport, mass disappearance, and creation.

Searching arXiv for the cited paper and closely related work to ground the article. The Hellinger discrepancy tensor denotes the Riemannian metric tensor associated with the local linearization of the Hellinger–Kantorovich (HK) distance at a reference measure $\mu_0$. In the terminology of "The Linearized Hellinger--Kantorovich Distance" (Cai et al., 2021), if “Hellinger discrepancy tensor” is not standard terminology, it precisely refers to this HK Riemannian metric tensor at $\mu_0$. The construction combines transport and mass variation in a single inner product, supports logarithmic and exponential maps, and yields a local Hilbertian embedding of measures around $\mu_0$. The resulting framework interpolates between Wasserstein-$2$ geometry and Fisher–Rao/Hellinger geometry through an intrinsic length scale $\kappa$, while retaining an exact static and dynamic interpretation of tangent directions, geodesics, and measure creation.

1. Geometric setting and dynamic formulation

The framework is posed on a domain $\Omega \subset \mathbb{R}^d$ that is convex, closed, bounded, and has non-empty interior. Lebesgue measure on $\Omega$ is denoted by $L$. The relevant spaces are signed Radon measures $\mathrm{meas}(\Omega)$, nonnegative Radon measures $\mathrm{meas}^+(\Omega)$, and absolutely continuous nonnegative reference measures $\mu_0$0 with respect to $\mu_0$1.

The HK distance is defined dynamically through a Benamou–Brenier-type action. For a time-dependent measure $\mu_0$2, momentum $\mu_0$3, and source $\mu_0$4, one imposes the continuity equation with source

$\mu_0$5

or, when $\mu_0$6 and $\mu_0$7,

$\mu_0$8

The canonical HK action is

$\mu_0$9

and the squared HK distance is the infimum of this action over admissible triples connecting $\mu_0$0 and $\mu_0$1 (Cai et al., 2021).

A scaled family replaces the factor $\mu_0$2 by $\mu_0$3: $\mu_0$4 The parameter $\mu_0$5 is the intrinsic length scale that balances transport and mass change. Precisely,

$\mu_0$6

For $\mu_0$7, transported mass never travels farther than distance $\mu_0$8, a global transport bound that later reappears as a cut-locus phenomenon.

2. Tangent representation and the metric tensor

Informally, the tangent space at $\mu_0$9 consists of pairs $2$0 with

$2$1

representing transport velocities and relative mass growth rates. For a smooth path $2$2 with $2$3, the infinitesimal constraint at $2$4 is

$2$5

together with a singular creation part at points where mass appears. Accordingly, the tangent representation contains a third component that encodes creation from nothing.

The logarithmic map is defined by

$2$6

where $2$7 is the singular “created” part and $2$8 are the initial transport and reaction components on the moving part. The induced inner product at $2$9 is

$\kappa$0

for any dominating measure $\kappa$1 with $\kappa$2. With scale parameter $\kappa$3, the term $\kappa$4 is replaced by $\kappa$5 (Cai et al., 2021).

This tensor measures transport by the $\kappa$6-weighted Euclidean norm of $\kappa$7 and mass change by the $\kappa$8-weighted Fisher–Rao norm of $\kappa$9, with the precise $\Omega \subset \mathbb{R}^d$0 or $\Omega \subset \mathbb{R}^d$1 factor fixed by the HK action. The associated local norm is

$\Omega \subset \mathbb{R}^d$2

again with $\Omega \subset \mathbb{R}^d$3 replaced by $\Omega \subset \mathbb{R}^d$4 in the scaled case. For a small displacement $\Omega \subset \mathbb{R}^d$5, the HK distance admits the first-order approximation

$\Omega \subset \mathbb{R}^d$6

When $\Omega \subset \mathbb{R}^d$7, the third term vanishes and the embedding is Hilbertian on $\Omega \subset \mathbb{R}^d$8.

3. Static formulation, tangent components, and exact norm identities

The same geometry has a static Kantorovich-type formulation with soft marginals: $\Omega \subset \mathbb{R}^d$9 with

$\Omega$0

and $\Omega$1 otherwise. If $\Omega$2 is nonnegative and absolutely continuous with respect to $\Omega$3, the minimizer is unique and is induced by a Monge map: $\Omega$4

Writing the Lebesgue decomposition with respect to the marginals $\Omega$5 as

$\Omega$6

one obtains explicit formulas for the tangent data on the moving part: $\Omega$7 with the convention $\Omega$8 if $\Omega$9, and

$L$0

On the disappearing part, $L$1-almost everywhere,

$L$2

while the remaining appearing mass is $L$3 (Cai et al., 2021).

These quantities satisfy the exact identity

$L$4

This identity is the exact counterpart of “distance equals squared Riemannian norm of the logarithmic map” and motivates the metric tensor. A plausible implication is that the tensor is not merely a heuristic quadratic surrogate: it is directly induced by the exact decomposition of an HK geodesic into moving, disappearing, and appearing components.

4. Logarithmic and exponential maps

The logarithmic map at $L$5 is defined from the optimal soft-marginal coupling by

$L$6

Along the constant-speed HK geodesic $L$7 from $L$8 to $L$9, it scales exactly as

$\mathrm{meas}(\Omega)$0

This linear scaling is the central structural fact underlying the local linearization.

The exponential map reconstructs a measure from tangent data. Given

$\mathrm{meas}(\Omega)$1

define

$\mathrm{meas}(\Omega)$2

$\mathrm{meas}(\Omega)$3

and choose $\mathrm{meas}(\Omega)$4 such that

$\mathrm{meas}(\Omega)$5

Then

$\mathrm{meas}(\Omega)$6

and

$\mathrm{meas}(\Omega)$7

The constant-speed geodesic is

$\mathrm{meas}(\Omega)$8

and this matches the geodesic constructed from the optimal soft-marginal coupling (Cai et al., 2021).

If $\mathrm{meas}(\Omega)$9 is nonnegative and absolutely continuous, the optimal soft-marginal coupling is unique and induced by a Monge map; hence the logarithmic map is unique. The exponential map reconstructs $\mathrm{meas}^+(\Omega)$0 exactly when fed the HK logarithm. This suggests that the local tensor, the logarithmic map, and the exponential map form a coherent Riemannian package, although the paper emphasizes local Euclidean behavior rather than a fully global smooth geometry.

5. Interpolation between Wasserstein and Fisher–Rao/Hellinger geometries

The scaled family $\mathrm{meas}^+(\Omega)$1 interpolates between transport-dominated and reaction-dominated regimes. As $\mathrm{meas}^+(\Omega)$2, transport dominates and $\mathrm{meas}^+(\Omega)$3 converges to $\mathrm{meas}^+(\Omega)$4. At the level of tensors, the HK metric reduces to

$\mathrm{meas}^+(\Omega)$5

which is the Wasserstein metric tensor at $\mathrm{meas}^+(\Omega)$6 on tangent velocities.

As $\mathrm{meas}^+(\Omega)$7, after dividing distances by $\mathrm{meas}^+(\Omega)$8, $\mathrm{meas}^+(\Omega)$9 converges to the Hellinger–Kakutani metric. Correspondingly, the tensor reduces to the Fisher–Rao inner product on reaction rates,

$\mu_0$00

plus the pure Hellinger term on measure creation, namely the square-root inner product on measures (Cai et al., 2021).

This places the Hellinger discrepancy tensor in a comparative triad:

$\mu_0$01

$\mu_0$02

and

$\mu_0$03

Transport and mass change are therefore measured in commensurate units. The paper states that this explains HK’s robustness to local mass fluctuations. A common misconception is to treat HK as merely a convex combination of Wasserstein and Hellinger metrics; the tensorial description shows instead that HK has its own Riemannian structure with an exact coupling of transport, relative reaction, and ex nihilo creation.

6. Geodesics, cut locus, and discrete embeddings

For Dirac masses, the geometry becomes explicit. If one connects $\mu_0$04 to $\mu_0$05 with $\mu_0$06, the HK geodesic remains a single moving Dirac,

$\mu_0$07

with

$\mu_0$08

and

$\mu_0$09

At $\mu_0$10,

$\mu_0$11

in agreement with the formulas for $\mu_0$12. When $\mu_0$13, transport switches off and mass teleports via a pure Fisher–Rao geodesic. The cut locus at $\mu_0$14 causes discontinuities in the logarithmic and exponential maps across that threshold; away from it, the Riemannian structure is smooth (Cai et al., 2021).

The same formalism yields a practical linearization pipeline for discrete data. For a discrete reference

$\mu_0$15

and samples $\mu_0$16, one solves the soft-marginal HK problem by entropic unbalanced Sinkhorn to obtain optimal plans $\mu_0$17. If $\mu_0$18 is approximated on a grid, barycentric projection yields an approximate Monge map $\mu_0$19 and disintegrations $\mu_0$20. One then computes

$\mu_0$21

followed by

$\mu_0$22

$\mu_0$23

with creation component $\mu_0$24. The linearized embedding of $\mu_0$25 is the vector field–scalar pair $\mu_0$26 equipped with inner product

$\mu_0$27

plus the Hellinger inner product of the creation parts if present.

The stated computational advantage is that embedding $\mu_0$28 samples requires $\mu_0$29 HK solves rather than $\mu_0$30 pairwise distances, after which PCA, LDA, and SVM can be applied in Euclidean space. The paper further notes that a wide-support $\mu_0$31, such as a uniform grid or HK barycenter, typically yields $\mu_0$32 and hence a purely Hilbert embedding. Accuracy is best when samples lie near $\mu_0$33 along HK geodesics; large displacements, especially those crossing the $\mu_0$34 cut locus, degrade linearization fidelity.

The Hellinger discrepancy tensor is therefore the local object that makes this pipeline possible: it converts HK geometry into a weighted Euclidean structure while preserving, to first order and in several cases exactly, the combined effects of transport, disappearance, and appearance of mass.