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REVIEW 3 major objections 5 minor 1 cited by

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Dataset distance runs near-linearly via sliced optimal transport

desk verdict The scaling idea and empirical results are real, but the metric proof is broken as stated: finite moments do not determine a distribution, so s-OTDD with k=5 is not a guaranteed metric. read the letter →

arxiv 2501.18901 v2 pith:YSD54K7X submitted 2025-01-31 cs.LG cs.AIstat.COstat.MEstat.ML

Khai Nguyen , Hai Nguyen , Tuan Pham , Nhat Ho This is my paper · ORCID
classification cs.LGcs.AIstat.COstat.MEstat.ML
keywords slicedoptimaltransportdatasetdistancemomenttransformprojectiontransferlearningdataaugmentationWassersteinmetricscalablecomparison
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces s-OTDD, a dataset-to-dataset distance that first projects every data point—feature plus label—down to a single number, turning each dataset into a one-dimensional distribution, and then measures the expected sliced Wasserstein distance between the two resulting distributions. The paper claims this quantity is a genuine metric on the space of feature-and-label distributions and that it can be evaluated in time $O(L(n\log n + dn))$ and memory $O(L(d+n))$, independent of the number of classes. If right, it gives practitioners a training-free, embedding-free way to compare datasets for transfer learning, data augmentation, or federated settings, while tracking the much costlier optimal transport dataset distance at scales where that baseline becomes infeasible.

What carries the argument

The load-bearing object is the data point projection $\mathrm{DP}^k$, built from a feature projection and $k$ Moment Transform Projections. For a label $q_y$, MTP first pushes the label's feature distribution onto a line through $FP_\theta$ and then records the scaled moment $\mathbb{E}[(FP_\theta(X))^\lambda]/\lambda!$, so the label becomes a vector of scalar moments. The data point projection is a weighted sum of one feature projection and $k$ such label moments, yielding one real number per data point; when the feature projection and MTP are injective, the push-forward of a dataset's empirical distribution through $\mathrm{DP}^k$ is injective, which is exactly what turns the expected sliced Wasserstein distance into a metric rather than just a dissimilarity score.

What would settle it

Build two discrete label distributions, realized as empirical class-conditional feature distributions, that share all projected moments of order 1 through 5 for every direction $\theta$ but differ at order 6, and pair them with identical feature distributions; then s-OTDD with $k=5$ would return zero even though the datasets differ, contradicting the identity-of-indiscernibles part of the metric claim.

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Extended reading notes

Core claim

The central claim is that a label, viewed as a distribution over feature vectors, can be compressed to a handful of scalars via Moment Transform Projection without losing the information needed for dataset comparison, and that combining those scalars with a feature projection maps each data point to a real number. The sliced optimal transport dataset distance is then the expected one-dimensional Wasserstein distance between the projected empirical distributions of two datasets, averaging over random projection directions, moment orders, and weights. The paper proves that s-OTDD is a valid metric on $\mathcal{P}(\mathcal{X} \times \mathcal{P}(\mathcal{X}))$ whenever this data-point projection is injective, and shows that the Monte Carlo estimate has error $O(L^{-1/2})$ with $O(L(n\log n + dn))$ time and $O(L(d+n))$ memory.

Load-bearing premise

The metric claim collapses if the moment-based label projection is not one-to-one, and the paper's injectivity proof requires moment conditions—infinitely many moments with an existing moment-generating function, or finitely many moments with positive-definite Hankel matrices and a growth bound—that are not verified on any real dataset.

Editorial extensions

If this is right

  • Both time and memory stay independent of the number of classes, so class imbalance and large label sets do not inflate the cost.
  • The one-dimensional Wasserstein computation uses sorting, so the distance can be computed in $O(Ln\log n)$ time after an $O(Lnd)$ projection pass, and each dataset's projections can be reused.
  • The Monte Carlo estimate converges at rate $O(L^{-1/2})$, so increasing the projection count predictably improves the approximation.
  • On MNIST and CIFAR10, s-OTDD's rankings correlate strongly with exact OTDD, and in transfer-learning and data-augmentation experiments it tracks performance gaps at a fraction of the runtime.
  • The construction handles disjoint label sets, because labels are represented as feature distributions rather than shared identifiers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The injectivity result suggests that any injective scalarization of the label distribution would yield a valid sliced dataset distance, so MTP is one instance of a broader recipe rather than the only possible choice.
  • Because the cost is independent of class count, the method is attractive for federated or continually growing label sets, though the paper does not run those experiments.
  • For practitioners, the finite-$k$ choice means s-OTDD should be treated as an ordinal ranking score unless the moment conditions are verified, since the formal metric guarantee is conditional on injectivity.
  • A direct testable extension would check whether s-OTDD rankings stay monotone under controlled label shifts, such as swapping labels or adding a rare class, against OTDD on the same dataset pairs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes sliced optimal transport dataset distance (s-OTDD), a training-free dataset distance. The key idea is to map each class label, viewed as a distribution over features, to a scalar through Moment Transform Projection (MTP), combine several such scalars with a feature projection to map each data point to a real number, and then define the dataset distance as the expected one-dimensional Wasserstein distance between the resulting projected empirical distributions over random projection parameters. The paper claims that s-OTDD is a valid metric on distributions over feature/label-distribution pairs, has near-linear time and memory complexity independent of the number of classes, and correlates well with exact OTDD, transfer-learning performance gaps, and augmentation accuracy. Experiments on MNIST, CIFAR10, NIST, text datasets, and Tiny-ImageNet support the empirical claims, and code is released. The main theoretical claim of metricity, however, is not supported as stated because the finite-moment injectivity argument in Proposition 1 is false and the proof of Proposition 2 contains an invalid zero-expectation-to-almost-sure step.

Significance. If the theoretical claims were correct, s-OTDD would be a valuable scalable alternative to OTDD, and the released code plus the near-linear complexity are genuine strengths. The empirical results, such as Spearman correlations around 0.90-0.93 with exact OTDD on MNIST/CIFAR10 subsets and transfer-learning correlations comparable to OTDD at lower cost, suggest practical utility independent of the metricity question. However, the central theoretical contribution is currently unsupported: finite moments do not identify distributions, and the passage from zero expectation to equality for all projection parameters is not justified. The paper needs a substantive correction of the metric claim, either by proving a correct version under appropriate assumptions (e.g., for finitely supported empirical measures with a sufficient number of moments) or by explicitly presenting s-OTDD as a pseudometric whose empirical utility is separate from metricity.

major comments (3)
  1. [Section 3.1, Proposition 1(2), and Appendix A.1] The finite-moment injectivity claim is false. Finitely many moments never determine a distribution, even when the truncated Hankel matrices are positive definite and the moment-growth bound holds; Hamburger uniqueness requires the full infinite moment sequence. A concrete counterexample is to take μ uniform on [0,1] and ν with Lebesgue density 1 + ε P_6^*(x), where P_6^* is the degree-6 shifted Legendre polynomial scaled to satisfy |P_6^*| ≤ 1 and ε ∈ (0,1). Then ν is a probability density, its first five moments agree with those of μ while the sixth moment differs, and for sufficiently small ε the finite Hankel matrices are positive definite and the stated growth bound holds. Hence MTP is not injective for finite Λ, the premise of Corollary 1 fails, and Proposition 2's identity-of-indiscernibles is false for such pairs. In particular, the λmax-capped version explicitly offered in Section 3.3 cannot be a metric, and the zero-truncated-Poisson implementation used in Section 4 is not covered by the finite-Λ proof (nor are the required MGF/injectivity conditions verified there).
  2. [Appendix A.2, proof of Proposition 2] The step 's-OTDD = 0 implies W_p^p(...) = 0 for all ψ, θ, λ, φ' is not justified. For a nonnegative random variable, zero expectation only implies almost-sure equality under the projection distribution; one cannot conclude equality for every parameter value without an additional argument such as continuity of the integrand together with full support, or an almost-sure injectivity statement. This is a second, independent gap in the metricity proof, and it remains even if MTP were injective.
  3. [Section 3.2, Corollary 1] The injectivity of the data point projection is asserted without a proof in the appendix, and it is load-bearing for the metric claim. Please provide a self-contained proof or a precise pointer to the exact theorem in Nguyen and Ho (2024), and verify that the convolution feature projection used for CIFAR10 and Tiny-ImageNet satisfies the required injectivity condition. The current dependence on an unstated assumption is not acceptable for the central theoretical contribution.
minor comments (5)
  1. [Definition 4 vs. Proposition 2] Definition 4 defines s-OTDD_p^p as an expectation, while Proposition 2 and its proof treat s-OTDD_p as the p-th root of that expectation; the notation should be aligned throughout.
  2. [Appendix A.2] In the triangle-inequality display, the first term on the right-hand side is written as W_p^p(DP PD3, DP PD2); it should be W_p^p(DP PD1, DP PD3).
  3. [Algorithm 1] The pseudocode for the data point projection uses ψ(k) where Definition 3 requires ψ(i′+1), and the loop 'for i = j to m' should be 'for j = 1 to m'; these typos make the algorithm inconsistent with the definition.
  4. [Appendix A.1, Proposition 1(2)] The displayed Hankel matrices contain entries m_{θ,μ,0} in row/column positions that do not follow the stated λmax pattern; the matrix display should be corrected.
  5. [Throughout] There are numerous typographical errors ('dicuss', 'Monte Caro', 'anaylsis', 'scability', 'Thabks') that should be cleaned up before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: s-OTDD is a new projection-based dataset distance whose metric claim is conditional on an attempted injectivity proof, not on fitted parameters or a self-citation loop.

full rationale

The s-OTDD (Definition 4) is defined directly as an expected W_p distance between one-dimensional projections of dataset empirical distributions; no constant is fitted to OTDD, transfer gaps, or augmentation accuracy, so the empirical correlations are genuine outputs. The metricity claim (Proposition 2) is conditional on injectivity of the data point projection, and the paper attempts to establish that injectivity via moment determinacy (Proposition 1) and a composition argument citing Nguyen & Ho (2024). The cited composition result is independent prior work with its own proof and does not assume the present target result, so it does not make the derivation circular. The finite-moment branch of Proposition 1 appears mathematically doubtful (finitely many moments never uniquely determine a distribution, so the k=5 setting may not yield a true metric), but that is a correctness risk rather than a circularity: the claim does not reduce by construction to its inputs. No self-definitional, fitted-input, or renamed-known-result step was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The s-OTDD introduces one new mathematical object, MTP, which is a map rather than an entity; no new physical or model entities are postulated. The main free choices are the moment count, moment-order sampling rates, projection count, and projection family, none of which is fitted to the target correlations but all of which affect reported performance.

free parameters (4)
  • Number of moments k = 5
    Fixed for all experiments; ablation in Figure 7 shows correlations are similar for k=2..6 on NIST, but k=5 is an arbitrary choice, not derived.
  • Truncated Poisson rate parameters r_1..r_k = 1,2,3,4,5
    Hyperparameters for sampling moment orders in Definition 4; no sensitivity analysis is reported.
  • Number of projections L = 10,000 (most experiments); 100,000 (augmentation); 500,000 (Tiny-ImageNet)
    Computational budget with direct effect on accuracy: Figure 3 shows correlations with the L=50,000 estimate grow with L; reported values depend on this choice.
  • Feature projection family = Radon for NIST/text; convolution for CIFAR/Tiny-ImageNet
    Chosen per dataset; Figure 12 shows convolution with 10 projections gives negative correlation (rho=-0.12) on NIST while Radon with 1,000 projections gives 0.39, so the choice materially affects reported correlations.
assumptions (5)
  • domain assumption Assumption 1: all projected scaled moments of the relevant distributions are finite.
    Required for MTP to be well-defined; invoked in Definition 2 and Proposition 1.
  • standard math The feature projection F P_theta is injective on the space of relevant distributions (e.g., Radon transform injectivity).
    Needed for MTP injectivity and hence for metricity; standard for Radon and sliced Wasserstein.
  • domain assumption Moment sequences uniquely determine the projected distribution: either MGFs exist (condition 1) or Hankel matrices are positive definite with growth condition (condition 2).
    Used in Proposition 1 to infer F P_theta sharp mu = F P_theta sharp nu from equal moments; these conditions are stated but not verified for empirical data.
  • standard math The data point projection inherits injectivity from MTP and feature projection via hierarchical hybrid projection (Nguyen and Ho, 2024).
    Corollary 1 relies on a theorem from a prior paper by the same research group; the present paper does not re-derive it.
  • domain assumption A label can be represented as the conditional distribution of features given that label, and dataset distance is computed on the joint distribution of features and label-distributions.
    Inherited from OTDD (Alvarez-Melis and Fusi, 2020); this is the modeling choice that makes MTP applicable.

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Cite this review

Pith. "Pith review of Lightspeed Geometric Dataset Distance via Sliced Optimal Transport." pith.science (2026). https://pith.science/paper/YSD54K7X

@misc{pith2026250118901,
  author       = {Pith},
  title        = {Pith review of: Lightspeed Geometric Dataset Distance via Sliced Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YSD54K7X}},
  note         = {Machine review of arXiv:2501.18901}
}
read the original abstract

We introduce sliced optimal transport dataset distance (s-OTDD), a model-agnostic, embedding-agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label sets. The core innovation is Moment Transform Projection (MTP), which maps a label, represented as a distribution over features, to a real number. Using MTP, we derive a data point projection that transforms datasets into one-dimensional distributions. The s-OTDD is defined as the expected Wasserstein distance between the projected distributions, with respect to random projection parameters. Leveraging the closed form solution of one-dimensional optimal transport, s-OTDD achieves (near-)linear computational complexity in the number of data points and feature dimensions and is independent of the number of classes. With its geometrically meaningful projection, s-OTDD strongly correlates with the optimal transport dataset distance while being more efficient than existing dataset discrepancy measures. Moreover, it correlates well with the performance gap in transfer learning and classification accuracy in data augmentation.

Figures

Figures reproduced from arXiv: 2501.18901 by the authors.

Figure 1
Figure 1. The figure shows distance correlation with OTDD (Exact) of OTDD (Gaussian approximation), WTE, CHSW, and s-OTDD. O(L(d+n)), where n is the number of data points and L is the number of projections. Interestingly, both time complex￾ity and memory complexity do not depend on the number of classes c and the maximum class size nmax (good for im￾balanced data). In a greater detail, the time complexity for applying project… view at source ↗
Figure 2
Figure 2. The figure shows computational time of OTDD (Exact), OTDD (Gaussian approx), WTE, CHSW (1,000, 5,000, 10,000 projections), and s-OTDD (1,000, 5,000, 10,000 projections) when varying size of two datasets. 0 10000 20000 30000 40000 50000 Number of Projections 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 P e a r s o n C o r r ela tio n ( ) Projections Analysis: MNIST Dataset Size 1,000 Dataset Size 5,000 Dataset Size 1… view at source ↗
Figure 3
Figure 3. The figure shows Pearson correlations of s-OTDD with s-OTDD (50,000 projections) when varying number of projections from 1,000 to 50,000 in MNIST dataset and CIFAR10 dataset. in Proposition 3. Furthermore, to analyze how the number of projections influences the correlations with OTDD, we select a specific dataset size and compare the s-OTDD re￾sults using fewer projections (500, 1,000, 5,000, and 10,000) against the… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The figure shows correlations of OTDD (Exact) and s-OTDD (10,000 projections) with the performance gap when conducting transfer learning in *NIST datasets. 150 175 200 225 250 275 300 325 OTDD (Exact) -10.0 0.0 10.0 20.0 30.0 40.0 50.0 Performance Gap (%) Distance vs A…
Figure 5
Figure 5. Figure 5: The figure shows correlations of OTDD (Exact) and s￾OTDD (10,000 projections) with the performance when conducting transfer learning in text datasets. PG serves a similar role to the Relative Drop metric in OTDD but offers a clearer view of adaptation transferability. …
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The figure shows correlations of OTDD (Exact) OTDD (Exact) and s-OTDD (100,000 projections) with the classification accuracy in data augmentation experiment [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: The figure shows distance correlations with OTDD (Exact) of s-OTDD when varying the number of projections. 0.5 1.0 1.5 2.0 CHSW (500 projections) ×10 3 260.0 270.0 280.0 290.0 300.0 310.0 OTDD (Exact) Distance Correlation: MNIST : 0.13 p-value: 7.27×10 1 0.0 1.0 2.0 3.…
Figure 10
Figure 10. Figure 10: The figure shows distance correlation with OTDD (Exact) of CHSW when varying the number of projections. improvement as the number of projections increases. This important observation suggests that s-OTDD in its population form is highly correlated with OTDD (Exact), a…
Figure 11
Figure 11. Figure 11: The figures show Pearson and Spearman correlation of OTDD (Exact), OTDD (Gaussian approximation), CHSW (10,000 projections), WTE, and s-OTDD (10,000 projections) with the performance gap when conducting transfer learning in *NIST datasets. 0.3 0.4 0.5 0.6 0.7 s-OTDD (…
Figure 12
Figure 12. Figure 12: The figure compares two projection methods, linear projections (Radon Transform projection) and convolution projection, using the *NIST dataset experiments. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 s-OTDD (5 moments - 10000 projections) 0.0 2.0 4.0 6.0 8.0 Performance Gap in F1 (%…
Figure 13
Figure 13. Figure 13: The figure shows Pearson and Spearman correlations of distance and Performance Gap in term of other metrics, i.e F1, Precision and Recall. C. Computational Devices For the runtime experiments and distance computations, we conducted tests using 8 CPU cores with 128GB o…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.