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arxiv: 2605.24329 · v1 · pith:YPGK4I6Inew · submitted 2026-05-23 · 🧮 math.MG · math.PR

On the Differential-Geometric Equivalence of Hellinger-Kantorovich and Cone-Wasserstein Spaces

Tristan Luca Saidi , Gonzalo Mena , Florian Gunsilius This is my paper

Pith reviewed 2026-06-30 12:38 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords Hellinger-Kantorovich distanceWasserstein spacecone constructionRiemannian geometryparallel transportgeodesic liftingnonnegative measures
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The pith

HK geometry on measures is locally isometric to Wasserstein geometry on their cone lifts along geodesics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the Hellinger-Kantorovich geometry can be locally matched to the Wasserstein geometry through a cone construction. It provides a method to lift certain HK geodesics to the cone such that the Riemannian structure is preserved at every point. This yields an isometry between the tangent spaces, allowing Wasserstein tools to compute HK quantities like parallel transport without solving complex PDEs. The authors also derive exact expressions for derivatives on cones and use simulations to show how mass and position interact in the geometry.

Core claim

We show that the cone representation of the HK geometry via the Wasserstein metric preserves the local Riemannian geometry along a class of lifted geodesics. Specifically, we give a constructive procedure that produces a Wasserstein geodesic on the cone along which the HK Riemannian geometry is preserved pointwise, yielding an explicit isometry of tangent spaces between HK geodesics and their Wasserstein lifts. This connection makes many Wasserstein-geometric tools available for HK computations.

What carries the argument

The constructive lifting procedure that maps HK geodesics to cone Wasserstein geodesics while preserving pointwise Riemannian geometry and giving tangent space isometries.

If this is right

  • Wasserstein parallel transport tools can approximate parallel transport on HK space by lifting to the cone.
  • Closed-form covariant derivative and parallel transport expressions hold on Euclidean metric cones from warped product theory.
  • Simulations reveal that HK geometry couples spatial and mass variations via the cone structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may reduce the computational cost of HK-based methods in applications with varying total mass.
  • Similar lifting techniques could connect other optimal transport variants to standard Wasserstein geometry.
  • The observed coupling might influence how mass variation is modeled in data analysis tasks.

Load-bearing premise

The pointwise preservation and isometry apply only to a specific class of lifted geodesics.

What would settle it

Observing a mismatch in the inner products of tangent vectors under the HK metric versus the cone Wasserstein metric at a point on one of the lifted geodesics would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.24329 by Florian Gunsilius, Gonzalo Mena, Tristan Luca Saidi.

Figure 1
Figure 1. Figure 1: Visualization of Hellinger-Kantorovich interpolation using the lifting procedure described in Algorithm 1, where grey lines denote radial coordinate lines. The measures µ0, µ1 consist of n0 = 1000 and n1 = 2000 draws from Gaussian probability measures with offset means. Note that we do not normalize the empirical measures, and thus µ0(R) ̸= µ1(R). Lifting Tangent Vector Fields. Fortunately, lifting tangent… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of Hellinger-Kantorovich interpolation using the lifting procedure described in Algorithm 2 under the same settings as in [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of parallel HK geodesics where µ1, µ2, µ3 consist of a variable number of samples from N(0, 2). We see that geodesics in (M+, HK) with parallel initial velocities experience mass change in the same direction (i.e. growth or shrinkage). In [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mass: 1000.00 Mass: 1000.00 Mass: 1000.00 Mass: 562.90 µ1 µ2 µ3 expµ1 (PTµ2→µ1 (logµ2 (µ3))) [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of parallel HK geodesics where µ1, µ2, µ3 consist of 1000 samples from Gaussian distributions with a fixed mean and differing variances. Parallel geodesics in (M+, HK) with parallel initial velocities experience similar changes in their second central moment. r p q γ up ∈ span{∂θ} aq bq ∂r PTC p→q (up) x [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Parallel transport on the cone visualized in polar coordinates, where x = θ. A purely spatial tangent vector up at p is parallel transported to a vector at q with both spatial and radial components. Thus, a purely spatial displacement may transport to a displacement that combines motion in space with mass variation. from spatial to radial variation [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualization of geodesic parallel transport on CΩ in polar coordinates (left) and cone coordinates (right). The visual consists of the parallel transport of v = 0.75∂r from (x0, r0) = (1.0, 0.5) to (x1, r1) = (3.0, 1.0). Appendix C. Proofs C.1. Proof of Proposition 3.3. We desire a z1 ∈ CΩ such that Z ′ (0+; z0, z1), the right derivative at 0, is equal to (vx, vr). Letting θ = ∥x1 − x0∥2, one can verify t… view at source ↗
read the original abstract

The Hellinger-Kantorovich (HK) space provides a natural geometry for nonnegative measures with varying total mass, but its differential-geometric structure is less well understood than that of the closely related Wasserstein space of probability measures. In this paper, we take a step toward resolving this issue. We show that the cone representation of the HK geometry via the Wasserstein metric preserves the local Riemannian geometry along a class of lifted geodesics. Specifically, we give a constructive procedure that produces a Wasserstein geodesic on the cone along which the HK Riemannian geometry is preserved pointwise, yielding an explicit isometry of tangent spaces between HK geodesics and their Wasserstein lifts. This connection makes many Wasserstein-geometric tools available for HK computations. Concretely, we use it to approximate parallel transport on HK space by lifting to the cone and applying recently developed Wasserstein parallel transport tools, circumventing the high-dimensional PDE arising from the HK covariant derivative. We also derive closed-form expressions for the covariant derivative and parallel transport on Euclidean metric cones, using the theory of warped-product manifolds. Finally, we present simulations illustrating the behavior of parallel geodesics in HK space, which reveal that the HK geometry couples spatial and mass variation through the geometry of the cone -- a feature with nontrivial implications for applied use of the framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that the cone representation of the Hellinger-Kantorovich (HK) geometry via the Wasserstein metric preserves the local Riemannian geometry along a class of lifted geodesics. It provides a constructive procedure to produce a Wasserstein geodesic on the cone that preserves the HK Riemannian geometry pointwise, resulting in an explicit isometry of tangent spaces. This is used to approximate parallel transport on HK space by lifting to the cone, and closed-form expressions for the covariant derivative and parallel transport on Euclidean metric cones are derived using warped-product manifold theory. Simulations of parallel geodesics in HK space are presented.

Significance. If the construction holds, the work provides a valuable bridge between HK and Wasserstein geometries, making Wasserstein tools available for HK computations and circumventing high-dimensional PDEs for parallel transport. The explicit isometry along the specified class, the closed-form expressions on cones via warped products, and the simulation insights into the coupling of spatial and mass variation are concrete strengths. The restriction to a specific class of lifted geodesics is stated clearly and avoids overclaiming global equivalence.

minor comments (3)
  1. [Abstract] The abstract and introduction refer to 'a class of lifted geodesics' without an immediate characterization of the class (e.g., via a defining property or equation); adding a one-sentence description would improve readability.
  2. [Section on warped-product manifolds] In the section deriving the closed-form covariant derivative on Euclidean metric cones, the warped-product formulas are invoked but the precise identification of the warping function with the cone metric is not written out; an explicit equation would aid verification.
  3. [Simulations section] The simulation figures show qualitative behavior of parallel geodesics but do not report quantitative error measures (e.g., deviation from the HK covariant derivative along the lift); including such metrics would strengthen the validation of the approximation procedure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the accurate summary of its contributions, and the recommendation of minor revision. The report correctly identifies the scope of the isometry result (restricted to the specified class of lifted geodesics) and the utility of the cone lift for approximating parallel transport.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a constructive geometric procedure that lifts HK geodesics to Wasserstein geodesics on the cone while preserving the Riemannian structure pointwise, yielding an explicit tangent-space isometry. This construction is derived from the cone representation and the theory of warped-product manifolds rather than from any fitted parameter, self-referential definition, or load-bearing self-citation. The abstract explicitly restricts the result to a specific class of lifted geodesics and does not claim global equivalence, so the central claim does not reduce to its own inputs by construction. No steps matching the enumerated circularity patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard differential geometry of Wasserstein and HK spaces plus the cone construction; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The HK space admits a Riemannian structure compatible with the cone embedding via the Wasserstein metric.
    Invoked when stating that the cone representation preserves local Riemannian geometry.
  • standard math Warped-product manifold theory applies directly to Euclidean metric cones.
    Used to derive closed-form covariant derivative and parallel transport.

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discussion (0)

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Reference graph

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7 extracted references · 6 canonical work pages · 2 internal anchors

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