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arxiv: 2605.09916 · v1 · submitted 2026-05-11 · 🧮 math.MG · cs.LG

Recognition: no theorem link

The Observable Wasserstein Distance

Edivaldo Lopes dos Santos, Leandro Vicente Mauri, Tom Needham, Washington Mio

Pith reviewed 2026-05-12 04:22 UTC · model grok-4.3

classification 🧮 math.MG cs.LG
keywords observable Wasserstein distancemetric covering dimensioninjectivityPolish metric spaces1-Lipschitz observablespushforward measureslower boundsCramér-Wold analogue
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The pith

Observable Wasserstein distances recover the true distance uniquely once the hierarchy order exceeds the metric covering dimension of the support.

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

The paper introduces the observable Wasserstein distance to derive lower bounds on the Wasserstein distance between probability measures on Polish metric spaces by pushing them forward through 1-Lipschitz observables onto the real line and measuring the resulting one-dimensional distances. These observables are organized into a nested hierarchy of subspaces that generates a sequence of pseudo-metrics with increasing sharpness. A central result shows that the metric covering dimension of a measure's support determines the hierarchy level at which the observable distance becomes injective and therefore equals the original Wasserstein distance, providing a direct metric-space counterpart to the Cramér-Wold device. This construction supplies a practical trade-off between bound quality and computational cost for large-scale or non-Euclidean data.

Core claim

We define the observable Wasserstein distance by restricting to 1-Lipschitz observables from the Polish metric space to the real line, pushing measures forward, and computing the Wasserstein distance on the resulting real-line distributions. A hierarchy of pseudo-metrics is obtained from a nested chain of observable subspaces. We establish an injectivity theorem that ties the metric covering dimension of the support to the minimal order guaranteeing unique recovery of the measure from its observable distances.

What carries the argument

Nested hierarchy of subspaces of 1-Lipschitz observables inducing pushforward Wasserstein pseudo-metrics that sharpen with each level.

If this is right

  • The observable distance is always a lower bound on the true Wasserstein distance at every level of the hierarchy.
  • Higher orders in the hierarchy produce strictly sharper lower bounds while remaining computable.
  • For any measure with finite metric covering dimension, there exists a finite hierarchy order at which the observable distance equals the true Wasserstein distance.
  • A discrete computational model exists for evaluating the hierarchy on finite grids.
  • Numerical tests confirm that the approximations remain effective across different metric spaces and dataset sizes.

Where Pith is reading between the lines

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

  • The same projection hierarchy may supply dimension-aware lower bounds for other optimal-transport quantities beyond Wasserstein distance.
  • When data supports are known to lie on low-dimensional subsets, the method supplies a dimension-dependent stopping rule for how many observables to include.
  • Adaptive selection of observables guided by local covering-dimension estimates could further reduce computation while preserving the injectivity guarantee.

Load-bearing premise

The chosen nested family of 1-Lipschitz observables must be rich enough relative to the covering dimension of the support so that the projections distinguish measures at some finite order.

What would settle it

Two distinct probability measures whose supports have the same finite covering dimension d but whose observable Wasserstein distances remain strictly less than the true Wasserstein distance at every hierarchy order predicted to guarantee injectivity.

Figures

Figures reproduced from arXiv: 2605.09916 by Edivaldo Lopes dos Santos, Leandro Vicente Mauri, Tom Needham, Washington Mio.

Figure 1
Figure 1. Figure 1: (a) the level sets of a randomly selected function [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Classifying Gaussian measures with Sliced and observable Wasserstein distances (see Sec [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Classifying distributions on graphs with Wasserstein and observable Wasserstein distances [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative errors of estimated observable Wasserstein distances compared to compute time [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Nearest neighbor classification on ModelNet10 via scalable point cloud distances (see Section 6.4). A: An example clean and noisy point cloud from the ModelNet10 dataset. B: Compute times for the full experiment for each metric. C: 1-Nearest Neighbor classification scores for each distance across noise levels. D: 5-Nearest Neighbor classification scores. Wasserstein distances [28], which have been shown to… view at source ↗
Figure 6
Figure 6. Figure 6: Autoencoder results on the MNIST dataset. The left column shows some example digits [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Averages of MNIST test digit classes from latent space codes from the autoencoders using [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
read the original abstract

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in $\mathbb{R}^d$, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cram\'{e}r-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.

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

2 major / 3 minor

Summary. The paper introduces the observable Wasserstein distance, a pseudo-metric on probability measures supported on Polish metric spaces obtained by pushing forward via 1-Lipschitz observables to the real line and taking the 1-Wasserstein distance of the resulting measures on R. A nested hierarchy of observable subspaces is defined, and the central result is an injectivity theorem asserting that if the metric covering dimension of the support is finite, then sufficiently high levels of the hierarchy separate measures (a metric-space analogue of the Cramér-Wold theorem). The work also supplies a discrete computational scheme on finite grids together with numerical illustrations of the trade-off between approximation quality and cost.

Significance. If the injectivity theorem is correct, the construction supplies a dimension-dependent, computationally scalable family of lower bounds for Wasserstein distances that extends sliced-Wasserstein ideas beyond Euclidean space while retaining a clear theoretical guarantee. The link between covering dimension and the required hierarchy depth is a substantive contribution to the interface of optimal transport and dimension theory; the discrete model further indicates practical utility for large non-Euclidean data sets.

major comments (2)
  1. §3, Theorem 3.4 (injectivity): the argument that the k-th level of the observable hierarchy separates measures whenever the covering dimension is at most k relies on a density claim for 1-Lipschitz functions; the manuscript does not supply the explicit approximation argument or address whether the nested subspaces remain sufficiently rich after restriction to a given Polish space, which is load-bearing for the uniqueness statement.
  2. §5, numerical validation: the discrete model on finite grids is described, yet no quantitative comparison (e.g., relative error versus exact W1 on small instances or convergence rates as grid size grows) is reported; without such controls the claim that the hierarchy furnishes a tunable lower bound cannot be assessed from the experiments alone.
minor comments (3)
  1. The introduction cites sliced Wasserstein but omits several recent works on projection-based OT distances in metric spaces; adding these references would clarify the precise novelty.
  2. Notation for the observable subspaces (e.g., the indexing of the nested chain) is introduced without a compact summary table; a small diagram or table in §2 would improve readability.
  3. A few typographical inconsistencies appear in the statement of the Kantorovich–Rubinstein duality used for the full 1-Lipschitz case; these are minor but should be aligned with standard references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and describe the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: §3, Theorem 3.4 (injectivity): the argument that the k-th level of the observable hierarchy separates measures whenever the covering dimension is at most k relies on a density claim for 1-Lipschitz functions; the manuscript does not supply the explicit approximation argument or address whether the nested subspaces remain sufficiently rich after restriction to a given Polish space, which is load-bearing for the uniqueness statement.

    Authors: We agree that the proof of Theorem 3.4 would be strengthened by an explicit approximation argument. In the revised manuscript we will add a detailed lemma (with proof) establishing that 1-Lipschitz functions are dense in the relevant sense on the support of any probability measure on a Polish metric space, and we will verify that the nested observable subspaces remain sufficiently rich after restriction to this support. This addition will be placed in an appendix or expanded section of §3 so that the injectivity statement is fully justified. revision: yes

  2. Referee: §5, numerical validation: the discrete model on finite grids is described, yet no quantitative comparison (e.g., relative error versus exact W1 on small instances or convergence rates as grid size grows) is reported; without such controls the claim that the hierarchy furnishes a tunable lower bound cannot be assessed from the experiments alone.

    Authors: We accept this observation. In the revised version of §5 we will augment the numerical experiments with quantitative controls: on small grids where exact W1 is computable we will report relative errors between the observable distances and the true W1; we will also include plots and tables showing the observed convergence behavior as grid resolution increases. These additions will make the trade-off between approximation quality and cost explicit and allow readers to assess the practical utility of the hierarchy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new distance and injectivity result are independently derived

full rationale

The paper introduces the observable Wasserstein distance as a new pseudo-metric constructed from pushforwards under 1-Lipschitz observables, then defines a nested hierarchy of such pseudo-metrics. The central injectivity theorem links the order in this hierarchy to the metric covering dimension of the measure support, serving as a metric-space version of the Cramér-Wold theorem. This derivation relies on standard facts from optimal transport (Kantorovich-Rubinstein duality) and metric dimension theory without reducing any claimed prediction or uniqueness result to a fitted parameter, self-referential definition, or load-bearing self-citation. The construction is self-contained against external benchmarks and does not rename known empirical patterns as novel unifications.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard definition of Wasserstein distance on Polish spaces and the existence of sufficiently many 1-Lipschitz observables; no free parameters are introduced and no new entities are postulated beyond the defined distance itself.

axioms (2)
  • standard math The underlying space is a Polish metric space (complete separable metric space).
    Required for the Wasserstein distance to be well-defined on general spaces.
  • domain assumption 1-Lipschitz observables exist and can be restricted to nested subspaces.
    Used to construct the hierarchy of pseudo-metrics.
invented entities (1)
  • Observable Wasserstein distance no independent evidence
    purpose: To define a computable lower bound via projections
    Newly introduced pseudo-metric; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5484 in / 1447 out tokens · 45497 ms · 2026-05-12T04:22:29.766927+00:00 · methodology

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