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arxiv: 2606.29665 · v1 · pith:6JAYFJDR · submitted 2026-06-29 · stat.ML · cs.LG

Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

Flor Martinez-Sermeno , Arturo Jaramillo , Johan Van Horebeek This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-30 04:54 UTCgrok-4.3pith:6JAYFJDRrecord.jsonopen to challenge →

classification stat.ML cs.LG
keywords Max-D-SWMax-Sliced Wasserstein distanceMultidimensional Scalingheavy-tailed distributionssample complexitymetric adjustmentembedding quality
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The pith

Max-D-SW aggregates sliced Wasserstein distances over orthonormal bases to improve MDS embeddings, especially for heavy-tailed data, while retaining comparable sample complexity.

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

The paper introduces Max-D-SW as an adjustment to the Max-Sliced Wasserstein distance for use inside Multidimensional Scaling. Where the original optimizes over single directions, Max-D-SW sums contributions across orthonormal bases. This produces visibly better low-dimensional embeddings on heavy-tailed distributions. Sample-complexity bounds stay at the same order as the max-sliced version. The authors also show that a metric with superior statistical rates need not deliver superior MDS results when used as input.

Core claim

Max-D-SW aggregates contributions over orthonormal bases rather than optimizing over single unit directions. This modification yields a clear numerical advantage in MDS outcomes, particularly for heavy-tailed distributions. Sample-complexity bounds remain statistically tractable and comparable to those of the max-sliced counterpart. Better sample complexity for a metric does not necessarily translate into better performance when that metric serves as input to MDS.

What carries the argument

Max-D-SW distance, formed by aggregating sliced Wasserstein distances over orthonormal bases instead of maximizing over single directions.

If this is right

  • MDS embeddings computed with Max-D-SW exhibit improved numerical outcomes relative to those using the max-sliced Wasserstein distance.
  • The improvement appears most clearly when the underlying data follow heavy-tailed distributions.
  • Sample-complexity rates for Max-D-SW match the order of the max-sliced version, keeping the method statistically tractable.
  • Superior sample complexity of a metric does not guarantee superior MDS performance when the metric is supplied as input.

Where Pith is reading between the lines

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

  • The aggregation step may capture directional information that single-direction maximization misses, offering a route to more stable embeddings when tails are heavy.
  • The observed gap between statistical rates and task performance suggests that metric design for visualization should be evaluated directly on the downstream embedding task rather than on general convergence bounds.
  • Similar base-aggregation adjustments could be tested on other sliced or projected distances used in nonlinear dimensionality reduction.

Load-bearing premise

Aggregating sliced Wasserstein distances over orthonormal bases produces a metric whose empirical behavior inside MDS is reliably superior for heavy-tailed data without introducing compensating distortions.

What would settle it

A controlled MDS experiment on heavy-tailed synthetic data in which Max-D-SW embeddings show equal or worse stress or visual quality than max-sliced Wasserstein embeddings would falsify the claimed numerical advantage.

Figures

Figures reproduced from arXiv: 2606.29665 by Arturo Jaramillo, Flor Martinez-Sermeno, Johan Van Horebeek.

Figure 1
Figure 1. Figure 1: Mean and variance of the distances between the reference Gaussian distribution µ and Gaussian distributions with increasing covariance parameter Si . the standard deviation from the mean of this distances. As expected, Max-Sliced Wasserstein fails to distinguish between the different distributions and assigns the same distance between µ and every νi . In this case, Sliced Wasserstein dis￾criminates correct… view at source ↗
Figure 2
Figure 2. Figure 2: Original images of the submarine and car, together with their corresponding edge-detected images. The result obtained through multidimensional scaling is presented in [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: First and second MDS coordinates (top row), and third and fourth MDS coordinates (bottom row), obtained using different distances. Acknowledgements Arturo Jaramillo Gil was supported by the grant CBF2023-2024-2088. References [1] Nicolas Bonneel, Julien Rabin, Gabriel Peyr´e, and Hanspeter Pfister. Sliced and radon wasserstein barycenters of measures, 01 2014. [2] Ingwer Borg and Patrick J. F. Groenen. Mod… view at source ↗
read the original abstract

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

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

1 major / 2 minor

Summary. The paper proposes Max-D-SW, a modification of the max-sliced Wasserstein distance that aggregates contributions over orthonormal bases rather than optimizing over single directions. It claims this yields a clear numerical advantage when used as a metric in multidimensional scaling (MDS), especially for heavy-tailed distributions, while establishing sample-complexity bounds comparable to the max-sliced version. The paper also observes that improved sample complexity does not necessarily imply better MDS performance.

Significance. If the numerical advantage and bounds hold, the work could offer a practical adjustment for MDS on non-light-tailed data and clarify the relationship between statistical rates and embedding quality. The explicit decoupling of sample complexity from MDS utility is a useful observation, but the heavy-tailed emphasis rests on empirical behavior whose connection to the stated tractability is not yet demonstrated.

major comments (1)
  1. [Abstract / sample-complexity section] Abstract and theoretical claims: the sample-complexity bounds are stated to remain 'comparable' to max-sliced Wasserstein, yet the highlighted application is to heavy-tailed distributions. Standard Wasserstein theory requires E[||X||^p]<∞ for the p-Wasserstein distance; if the proof of the Max-D-SW bound invokes this moment condition (as is typical), the bound cannot be invoked for the very regime where the numerical advantage is claimed. This makes the tractability statement load-bearing for the central empirical claim.
minor comments (2)
  1. [Introduction] The abstract refers to 'Max-D-SW' without an explicit definition or equation; a short displayed equation or pseudocode in the introduction would clarify the aggregation over orthonormal bases versus single-direction optimization.
  2. [Method] No mention of how the orthonormal bases are chosen or whether the aggregation is normalized; this detail affects both the metric property and computational cost.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important clarification needed regarding the moment conditions in our sample-complexity analysis. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / sample-complexity section] Abstract and theoretical claims: the sample-complexity bounds are stated to remain 'comparable' to max-sliced Wasserstein, yet the highlighted application is to heavy-tailed distributions. Standard Wasserstein theory requires E[||X||^p]<∞ for the p-Wasserstein distance; if the proof of the Max-D-SW bound invokes this moment condition (as is typical), the bound cannot be invoked for the very regime where the numerical advantage is claimed. This makes the tractability statement load-bearing for the central empirical claim.

    Authors: We agree that the sample-complexity bounds for Max-D-SW, like those for max-sliced Wasserstein, rely on the standard assumption E[||X||^p] < ∞. Our proof follows the same moment condition as the baseline. The numerical experiments demonstrating advantages on heavy-tailed data were performed on distributions satisfying this condition (e.g., multivariate Student's t with degrees of freedom chosen to ensure finite moments while retaining heavy tails). We will revise the abstract, introduction, and theoretical sections to explicitly state the moment assumptions and to clarify that the claimed numerical gains are shown under these conditions. This removes any ambiguity about the applicability of the bounds to the reported experiments. revision: yes

Circularity Check

0 steps flagged

No circularity; abstract states claims without equations or derivations

full rationale

The provided abstract and context contain no equations, no fitting procedures, no self-citations, and no derivation chain. Claims of numerical advantage in MDS and sample-complexity bounds are asserted but not derived or shown, so no reduction to inputs by construction is possible. The skeptic note concerns moment conditions for heavy-tailed data (a correctness issue) rather than circularity. This matches the default expectation of no significant circularity when no load-bearing steps are inspectable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, background axioms, or new entities; ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5665 in / 1073 out tokens · 35651 ms · 2026-06-30T04:54:42.263896+00:00 · methodology

discussion (0)

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Reference graph

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