arxiv: 2605.00375 · v1 · submitted 2026-05-01 · 🧮 math.FA · math.PR

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

Fatma Terzioglu, Ryan Murray

Pith reviewed 2026-05-09 19:02 UTC · model grok-4.3

classification 🧮 math.FA math.PR

keywords k-plane transformWasserstein distancestability estimatesSobolev normRadon measuresFourier metricsliced WassersteinHölder continuity

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The pith

For absolutely continuous compactly supported probability measures with bounded densities, the 2-Wasserstein distance is equivalent to the (k/2-1)-order Sobolev norm of the k-plane data of the difference of their densities.

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

The paper establishes stability estimates for the k-plane transform on positive Radon measures, first by introducing a metric on the k-plane data that is bi-Lipschitz equivalent to a generalized Fourier metric accounting for barycenter and total mass differences. It then extends Hölder-type comparisons between Fourier and Wasserstein metrics to positive Radon measures under uniform bounds on centered moments of order slightly higher than 2, yielding Hölder stability in the 2-Wasserstein distance for centered probabilities. The work also proves a two-sided Hölder comparison between the 2-Wasserstein distance and its max-sliced analogue under the same moment conditions, extending the comparison to general positive Radon measures by separating barycenter and mass terms. The culminating result is a strong equivalence, for absolutely continuous compactly supported measures with bounded densities, between the 2-Wasserstein distance and the Sobolev norm of order k/2-1 applied to the k-plane data of the density difference.

Core claim

The central claim is that the k-plane transform on measures admits stability estimates in Fourier and Wasserstein metrics, with the strongest form being an equivalence: for absolutely continuous compactly supported probability measures with bounded densities, the 2-Wasserstein distance between two such measures equals (up to constants) the (k/2-1)-order Sobolev norm of the k-plane transform of the difference of their densities. This equivalence is obtained after first establishing bi-Lipschitz stability of a metric on k-plane data with a generalized Fourier metric, deriving Hölder comparisons between Wasserstein and Fourier metrics for Radon measures, and proving two-sided Hölder bounds for

What carries the argument

The k-plane transform, which maps a measure to its integrals over all k-dimensional affine planes, together with associated metrics on the resulting data that are shown equivalent to Fourier and Wasserstein distances on the original measures.

If this is right

The k-plane transform is stable with respect to Wasserstein distance for measures satisfying the stated regularity.
Max-sliced Wasserstein distance is Hölder equivalent to full 2-Wasserstein distance under uniform higher-moment bounds.
Generalized distances that separate barycenter and mass allow the stability results to extend from centered probability measures to arbitrary positive Radon measures.
Hölder comparisons between Fourier and Wasserstein metrics hold for positive Radon measures when centered moments of order slightly above 2 are uniformly bounded.

Where Pith is reading between the lines

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

The equivalence suggests that k-plane data may be sufficient to control optimal transport distances in inverse problems with sufficient smoothness.
The moment bounds required for the Hölder comparisons indicate that the results may fail for measures with heavy tails, pointing to a possible need for truncation or regularization in applications.
Connections to tomographic reconstruction follow naturally, where controlling a Sobolev norm on projected data would bound the Wasserstein error between original measures.
In computational settings the max-sliced Wasserstein distance could serve as an efficient proxy whose error relative to full Wasserstein distance is explicitly controlled by the moment assumptions.

Load-bearing premise

The measures are absolutely continuous with respect to Lebesgue measure, compactly supported, and possess bounded densities.

What would settle it

Two distinct absolutely continuous compactly supported probability measures with bounded densities that produce identical k-plane data yet have different 2-Wasserstein distances would falsify the claimed equivalence.

read the original 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.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit stability estimates for the k-plane transform on Radon measures that link to generalized Fourier and Wasserstein metrics, plus a two-sided Hölder comparison for max-sliced W2, but the final Sobolev equivalence claim carries an exponent that ignores ambient dimension and needs verification.

read the letter

The core new pieces are a bi-Lipschitz equivalence between a metric on k-plane data and a generalized Fourier distance that separates barycenter, mass, and centered normalized parts, plus an extension of the Carrillo-Toscani Hölder comparison to max-sliced Wasserstein under moment bounds slightly above order 2. These look like concrete additions to the literature on integral transforms and optimal transport distances for measures. The work also gives a Hölder stability result for the k-plane transform itself in terms of a generalized W2 distance, and for centered probabilities it recovers a Hölder bound directly in W2. That chain of estimates is the part that could be useful to people who need quantitative control on how much the k-plane data determines the underlying measure. The proofs appear to rest on standard measure theory and the cited prior comparison, without obvious circularity or free parameters. The final statement, however, asserts a strong equivalence between W2 and the H^{k/2-1} norm of the k-plane data of the density difference for absolutely continuous compactly supported probabilities with bounded densities. Standard Fourier analysis of the k-plane transform produces a Sobolev shift of order (d-k)/2, so an inverse estimate consistent with W2 scaling would normally place the data norm in a space whose index depends on both k and d. The claimed exponent k/2-1 has no d in it, which only works if d equals k. The abstract and the stress-test note do not show how the earlier Fourier or Hölder steps cancel this dependence, so the claim as stated looks like it may contain a gap. If the full proofs contain a different argument or a hidden d-dependent factor, that needs to be made explicit. This paper is aimed at specialists in functional analysis and optimal transport who work with Radon measures, integral transforms, and sliced distances. A reader already familiar with Carrillo-Toscani and the basic properties of the k-plane transform will get the most out of the explicit constants and the max-sliced comparison. The results are narrow enough that they do not demand broad attention, but the estimates are specific enough to merit checking. I would send it to peer review so that the derivations, moment assumptions, and the Sobolev exponent can be examined in detail.

Referee Report

1 major / 2 minor

Summary. The paper proves stability estimates for the k-plane transform acting on positive Radon measures. It introduces a metric on k-plane data and shows bi-Lipschitz equivalence to a generalized Fourier metric that accounts for differences in centered normalized measures, barycenters, and total mass. Building on the Carrillo-Toscani Hölder comparison between Fourier and Wasserstein metrics, it derives an analogous Hölder stability result for the k-plane transform in terms of a generalized 2-Wasserstein distance, with a special case for centered probabilities. It further establishes a two-sided Hölder comparison between W_2 and max-sliced W_2 under moment bounds, extends it to general Radon measures, and concludes with a strong equivalence, for absolutely continuous compactly supported probabilities with bounded densities, between W_2(μ,ν) and the H^{k/2-1} Sobolev norm of the k-plane transform of ρ_μ - ρ_ν.

Significance. If the central claims hold, the work supplies concrete stability and comparison results linking the k-plane transform to Wasserstein and Fourier metrics on measures. The bi-Lipschitz Fourier-metric equivalence and the extension of the Carrillo-Toscani Hölder lift to the k-plane setting are technically useful; the final Sobolev equivalence, if correct, would give a direct link between optimal-transport distance and integral-geometry data. The moment assumptions are standard and the results are stated for both probabilities and general Radon measures.

major comments (1)

Final claim (abstract and §6): the asserted strong equivalence between W_2(μ,ν) and the H^{k/2-1} norm of the k-plane data of ρ_μ - ρ_ν is dimensionally inconsistent with the standard Fourier-slice analysis of the k-plane transform. The transform induces a Sobolev shift of order (d-k)/2 (modulo angular regularity on the Grassmannian). For densities, W_2 control is comparable to an H^{-1}-type norm; inverting the shift therefore places the data norm in H^{-1-(d-k)/2}. Equating the exponents k/2-1 = -1-(d-k)/2 forces d=k, which is false for k<d. The earlier Fourier-metric stability and Carrillo-Toscani lift do not cancel this d-dependence, so the specific exponent requires explicit justification or correction.

minor comments (2)

Notation: the generalized Fourier metric and the precise definition of the k-plane data metric should be displayed as numbered equations rather than inline text.
Assumptions: the precise moment order required for the Hölder comparisons (slightly higher than 2) should be stated explicitly with the constant appearing in the exponent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying a potential inconsistency in the exponent appearing in our final Sobolev equivalence result. We address this point below and will incorporate the necessary correction and clarification in the revised manuscript.

read point-by-point responses

Referee: Final claim (abstract and §6): the asserted strong equivalence between W_2(μ,ν) and the H^{k/2-1} norm of the k-plane data of ρ_μ - ρ_ν is dimensionally inconsistent with the standard Fourier-slice analysis of the k-plane transform. The transform induces a Sobolev shift of order (d-k)/2 (modulo angular regularity on the Grassmannian). For densities, W_2 control is comparable to an H^{-1}-type norm; inverting the shift therefore places the data norm in H^{-1-(d-k)/2}. Equating the exponents k/2-1 = -1-(d-k)/2 forces d=k, which is false for k<d. The earlier Fourier-metric stability and Carrillo-Toscani lift do not cancel this d-dependence, so the specific exponent requires explicit justification or correction.

Authors: We agree with the referee that the claimed exponent k/2-1 is inconsistent with the Sobolev mapping properties of the k-plane transform under the Fourier-slice theorem. The bi-Lipschitz equivalence to the generalized Fourier metric and the subsequent Hölder lift to Wasserstein distance do not automatically absorb the (d-k)/2 shift induced by the transform on the data side. We will revise Section 6 (and the abstract) to replace the exponent with the corrected value -1-(d-k)/2, add an explicit paragraph deriving the shift from the Fourier-slice relation (including the angular regularity on the Grassmannian), and verify that the resulting two-sided estimate remains valid under the stated assumptions of compact support and bounded densities. The earlier stability results are unaffected. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain builds stability estimates for the k-plane transform by first introducing a metric on k-plane data and proving bi-Lipschitz equivalence to a generalized Fourier metric, then extending an external Hölder comparison from Carrillo-Toscani to obtain Hölder stability in Wasserstein distance, and finally deriving the Sobolev equivalence for AC measures as a consequence. These steps rely on standard Fourier analysis, measure theory, and the cited external result rather than self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The central claims retain independent content and do not reduce to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard properties of Radon measures, Fourier transforms, and Wasserstein metrics from prior literature without introducing new free parameters or postulated entities.

axioms (2)

standard math Standard properties of positive Radon measures, their Fourier transforms, and centering/normalization operations
Invoked for the definition of the generalized Fourier metric and stability estimates.
standard math Existence and basic inequalities for 2-Wasserstein distance and its sliced variants
Basis for the Hölder comparisons and extensions to non-probability measures.

pith-pipeline@v0.9.0 · 5631 in / 1380 out tokens · 46233 ms · 2026-05-09T19:02:54.952440+00:00 · methodology

discussion (0)

Reference graph

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