Stability Estimates for the $k$-plane Transform on Measures and a Hölder-Type Comparison Between Wasserstein and Max-Sliced Wasserstein Distances

Abstract: We establish stability estimates for the $k$-plane transform on positive Radon measures, with particular emphasis on Fourier and Wasserstein metrics. We first introduce a metric on $k$-plane data and prove a bi-Lipschitz stability estimate showing that this metric is equivalent to a generalized Fourier metric obtained by combining the $d_2$-distance between centered normalized measures with separate terms accounting for differences in barycenter and total mass. Next, building on a Hölder-type comparison between Fourier and Wasserstein metrics due to Carrillo and Toscani, we prove an analogous estimate for positive Radon measures under uniform bounds on centered moments of order slightly higher than $2$. As a consequence, we obtain a Hölder-type stability estimate for the $k$-plane transform in terms of a generalized $2$-Wasserstein distance. For centered probability measures, this yields a Hölder stability estimate in the $2$-Wasserstein distance $W_2$. We also study the relation between $W_2$ and its max-sliced analogue. For centered probability measures with uniformly bounded moments of order slightly higher than $2$, we prove a two-sided Hölder-type comparison between $W_2$ and max-sliced $W_2$. We then extend this comparison to positive Radon measures by combining the corresponding estimate for centered normalized measures with separate terms accounting for differences in barycenter and total mass. Finally, for absolutely continuous compactly supported probability measures with bounded densities, we obtain a strong equivalence between the $2$-Wasserstein distance of the measures and the $(k/2-1)$-order Sobolev norm of the $k$-plane data of the difference of their densities.

• The paper introduces a bi-Lipschitz metric that quantitatively links k-plane projections to augmented Fourier distances for measure stability.
• The paper establishes a Hölder-type estimate that compares generalized Fourier and Wasserstein distances under higher moment controls.
• The paper demonstrates a sharp two-sided equivalence between Wasserstein and max-sliced Wasserstein distances, with implications for inverse problems in tomography.

Stability Estimates for the $k$-Plane Transform on Measures and Hölder-Type Comparison of Wasserstein and Max-Sliced Wasserstein Distances

Overview and Motivation

This paper addresses the stability analysis of the $k$-plane transform in the context of measure-theoretic and optimal transport frameworks, focusing on its behavior under Fourier and Wasserstein metrics. The study is motivated by the central role of the $k$-plane transform in integral geometry, tomography, and data science, especially as it relates to projection-based distances (such as sliced, max-sliced, and subspace-robust Wasserstein metrics) that are computationally advantageous and practically relevant.

The main objective is to extend and quantitatively refine classical stability results for function spaces (notably Sobolev stability) to the more general setting of positive Radon measures. The work delineates connections between various generalized distances—Fourier-based and Wasserstein-type—when applied to the transform and offers sharp, quantitative estimates, including bi-Lipschitz and Hölder-type comparisons.

Main Contributions

1. Bi-Lipschitz Metric Equivalence for $k$-Plane Data

A novel metric $D(P\mu, P\nu)$ is introduced, defined as the supremum over the Grassmannian of generalized Fourier distances between projections of measures. The core result establishes a bi-Lipschitz equivalence between $D(P\mu, P\nu)$ and an augmented Fourier metric $\widetilde d_2(\mu, \nu)$, which aggregates:

• the $d_2$-Fourier distance between centered, normalized components,
• absolute differences in barycenters,
• and total mass discrepancies.

For centered probability measures, this metric reduces exactly to the standard $d_2$-Fourier metric. This equivalence supports robust quantitative stability for the $k$-plane transform on measures, mirroring the regularity gains known from Sobolev mapping theory.

2. Hölder-Type Comparison of Fourier and Wasserstein Metrics

Building on the Carrillo–Toscani framework, an analogous Hölder-type estimate between the generalized Fourier and Wasserstein distances is established for positive Radon measures. Specifically, under moment control of order slightly above $k$0, the following structural estimate holds:

$k$1

with constants $k$2 and $k$3 depending on the moment bounds and space dimension. This generalizes earlier results for probability measures and allows for explicit control over barycenter and mass fluctuations.

3. Two-Sided Hölder Comparisons: Wasserstein vs Max-Sliced Wasserstein

The paper analyzes the relationship between the $k$4-Wasserstein metric and its max-sliced version:

$k$5

For centered probability measures with sufficiently high bounded moments, a sharp two-sided Hölder-type inequality is proven:

$k$6

with the extension to positive Radon measures achieved through coordinated management of barycenter and mass terms.

4. Bi-Lipschitz Equivalence for Absolutely Continuous Compactly Supported Measures

For absolutely continuous, compactly supported probability measures $k$7, $k$8 with densities bounded above and below, it is shown that:

$k$9

i.e., the $k$0-plane data norm in a specific Sobolev space equivalently quantifies the $k$1-Wasserstein distance, up to constants dependent on density bounds and support geometry.

Technical Innovations

A principal technical device is the generalization of Fourier and Wasserstein distances to measures of arbitrary finite mass and barycenter, formalized through the construction of $k$2 and $k$3. The authors leverage properties of orthogonal projections, barycenters, and pushforwards, together with the Fourier–slice theorem for measures, to translate stability and equivalence results across function space, measure space, and projection-based frameworks.

Uniform control of higher-order moments is used to transfer estimates from measures to their projected versions and to ensure the applicability of Hölder-type inequalities.

Numerical and Theoretical Implications

The main numerical implications lie in the explicit two-sided control of the stability of inverse problems based on $k$4-plane projection data, relevant for computed tomography and other imaging modalities. The measure-level formulation is better suited for sparse, singular, or particle-type data typical in practical scenarios and underpins emerging applications in data analysis using optimal transport.

The equivalence and stability results for projection-based metrics (e.g., max-sliced $k$5) are significant for scalable computational optimal transport, aligning with current trends in high-dimensional data science and machine learning, where full Wasserstein computations are prohibitive.

Future Directions

The results suggest several further research directions:

• Refinement of the Hölder exponents and extension to measures with weaker moment or support conditions.
• Generalization to broader integral transforms beyond the $k$6-plane case, including tensor or nonlinear analogues.
• Investigation of the stability and equivalence properties for signed measures or distributions, with applications in broader inverse and imaging problems.
• Exploitation of these stability results in designing new regularization schemes for inverse problems, especially those utilizing unbalanced optimal transport metrics.

Conclusion

This work systematically extends the theory of the $k$7-plane transform to the metric space of measures, providing bi-Lipschitz-type and Hölder-type stability results in both Fourier and Wasserstein frameworks. The methodology offers a bridge between classic integral geometry and modern optimal transport, with both theoretical significance and practical relevance for tomographic and machine learning applications. The results lay a robust foundation for future explorations of geometric transforms in measure-theoretic and computational contexts.

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Reference: "Stability Estimates for the $k$8-plane Transform on Measures and a Hölder-Type Comparison Between Wasserstein and Max-Sliced Wasserstein Distances" (2605.00375)