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

TL;DR

This paper establishes stability estimates for the $k$-plane transform on measures, linking Fourier, Wasserstein, and max-sliced Wasserstein metrics, and introduces new bounds and equivalences for these distances.

Contribution

It provides novel stability estimates connecting the $k$-plane transform with Fourier and Wasserstein metrics, including H"older-type bounds and relations between Wasserstein variants.

Findings

Bi-Lipschitz stability estimate for $k$-plane data metric.

H"older-type stability estimate for the $k$-plane transform in Wasserstein distance.

Two-sided H"older comparison between $W_2$ and max-sliced $W_2$.

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 H\"older-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 H\"older-type stability estimate for the $k$-plane transform in terms of a generalized $2$-Wasserstein distance. For centered probability measures, this yields a H\"older 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 H\"older-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.