Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices

Ashkan Shahbazi; Huy Tran; John R. Hershey; Soheil Kolouri; Yikun Bai

REVIEW 3 major objections 6 minor 1 cited by

Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper argues that classical Sliced-Wasserstein, with a properly tuned learning rate, can match or beat more complex informative-slice variants in common learning tasks.

desk verdict Solid subspace scaling theorem marred by an overstated learning-rate absorption claim and a false appendix proposition; still worth refereeing. read the letter →

arxiv 2411.10651 v1 pith:ZQ7TCR3I submitted 2024-11-16 cs.LG cs.AIcs.CVstat.APstat.COstat.ML

Huy Tran , Yikun Bai , Ashkan Shahbazi , John R. Hershey , Soheil Kolouri This is my paper · ORCID

classification cs.LGcs.AIcs.CVstat.APstat.COstat.ML

keywords Sliced-Wassersteindistanceinformativesliceslearningratelow-dimensionalsubspaceoptimaltransportgradientflowgenerativemodeling

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

The paper tries to establish that the standard Sliced-Wasserstein distance (SWD), often dismissed in high dimensions because random projections are uninformative, already contains the information those projections miss: when two distributions live in a common low-dimensional subspace, the ambient SWD is exactly a fixed scalar multiple of the SWD computed inside that subspace. Because the scalar depends only on ambient and effective dimensions, reweighting every slice by its informativeness reduces to multiplying the whole distance by one number, which in gradient-based learning is absorbed by the standard learning-rate search. The authors propose a $\phi$-weighting framework that unifies classical SWD with Max-SW, energy-based, Markovian, and random-path variants, then show empirically across gradient flow, color transfer, and generative modeling on FFHQ latents that a properly configured classical SWD often matches or surpasses these variants. A sympathetic reader would care because it means much of the added complexity of informative-slice SW variants may be unnecessary for common tasks: the plain distance plus a tuned step size can be both simpler and cheaper.

What carries the argument

The load-bearing object is the $\phi$-weighting formulation $gSW_p^p(\mu,\nu;\sigma,\rho_\phi) = \int_{S^{d-1}} \rho_\phi(\phi(\theta))\, W_p^p(\theta_\#\mu, \theta_\#\nu)\, d\sigma(\theta)$, which reweights each projected 1D Wasserstein by a function of slice informativeness instead of resampling slices. With the ES-aligned informativeness $\phi_U(\theta) = \lVert U^\top \theta \rVert$ and the reciprocal weighting $\rho_\phi(r) = r^{-p}$, Proposition 4.7 shows each slice contribution satisfies $W_p^p(\theta_\#\mu_d, \theta_\#\nu_d) = \lVert U^\top \theta \rVert^p W_p^p(\theta_{k\#}\mu_k, \theta_{k\#}\nu_k)$, so the reweighted distance equals the SWD in the $k$-dimensional effective subspace. Theorem 4.9 then identifies the expectation of $\lVert U^\top \theta \rVert^p$ with the ratio $C_k/C_d$ of two Gamma-function constants, turning per-slice rescaling into one global scalar. Proposition 4.10 extends this to finite samples and Monte Carlo slice estimators, with variance $O(1/L)$. The machinery's role is to convert the concentration-of-measure objection, that most random slices are nearly orthogonal to the data subspace, into a harmless multiplicative constant that learning-rate tuning already handles.

What would settle it

One concrete check: take two distributions on a common $k$-dimensional subspace and verify the empirical ratio of ambient to subspace sliced-Wasserstein approaches $C_k/C_d$ as the number of slices grows; the broader learning claim would be refuted by a common task, after a full learning-rate sweep, where plain SWD falls materially short of a tuned informative-slice variant.

Watch

Extended reading notes

Core claim

Under Assumption 4.1, where both measures are supported on the same $k$-dimensional linear subspace $V_k$ of $\mathbb{R}^d$, the $p$-th power of the sliced-Wasserstein distance obeys $SW_p^p(\mu_d, \nu_d) = (C_k/C_d)\, SW_p^p(\mu_k, \nu_k)$, with $C_d = 2^{p/2}\Gamma((d+p)/2)/\Gamma(d/2)$ and $C_k$ defined analogously. In other words, the information lost by averaging over all random projections of high-dimensional data is not a per-slice accident: it is a single dimension-dependent scaling constant. Defining slice informativeness as $\phi_U(\theta) = \lVert U^\top \theta \rVert$, the alignment of a direction $\theta$ with the effective subspace, rescaling each one-dimensional Wasserstein contribution by $\phi_U(\theta)^{-p}$ exactly cancels the implicit downweighting of misaligned slices, and in expectation this per-slice reweighting collapses to the global factor $C_d/C_k$. The paper's central claim is that in gradient-based learning this global factor is absorbed by learning-rate selection, so classical SWD with a properly chosen step size obtains informative slices for free and can perform competitively with more complex SW variants in common learning tasks.

Load-bearing premise

The argument assumes the two distributions sit on the same low-dimensional linear subspace, and then presumes the resulting rescaling factor stays roughly constant across minibatches so a single learning rate can absorb it.

Editorial extensions

If this is right

If the central claim is correct, the classical SWD needs no data-dependent slicing distribution to be competitive in common tasks: a learning-rate sweep over the plain distance is sufficient.
The effective-dimensionality dependence of the scaling factor means the same classical SWD automatically adapts to data with lower intrinsic dimension, without explicitly estimating the subspace.
Nearly all informative-slice SW variants, including Max-SW, EBSW, Markovian SW, and RPSW, are recovered as special cases of the $\phi$-weighting framework, so the metric properties of the classical distance are preserved when the global rescaling is applied.
Because the global factor can be absorbed into a learning rate, comparisons among SW variants in learning tasks should control for learning-rate search; reported gaps can shrink or reverse when the classical baseline is tuned.
Proposition 4.10 gives convergence rates for the gap between the ambient Monte Carlo SWD and the scaled subspace SWD, so users know how many slices are needed for the scalar-factor reduction to hold.

Reading between the lines

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

A testable extension is to compare classical SWD against informative-slice variants on data whose support is a curved manifold rather than a linear subspace, where per-slice reweighting may not collapse to a global scalar and the advantage of adaptive slicing could reappear.
The assumption of a shared common subspace for both distributions is stronger than the manifold hypothesis; if source and target occupy different low-dimensional subspaces, the global-factor argument may fail, and explicit subspace-aware methods would retain an edge.
Because the scaling factor depends only on ambient and effective dimension, one could estimate the effective dimension from minibatches and predict the optimal learning-rate shift across tasks, making the learning-rate search more principled.
If the claim is right, benchmark protocol changes: future SW-variant evaluations should include a well-tuned classical SWD baseline, otherwise apparent gains may reflect learning-rate mismatch rather than a genuine advantage of the new method.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: read this if you work on sliced Wasserstein. It does something simple and useful—it shows that under a linear subspace support assumption, the classic SWD is a fixed scalar multiple of the SWD computed in that subspace—and it backs the theory with an honest, broad set of experiments. But the identity is exact only for the population SWD. The step from there to "absorb the scaling into a fixed learning rate, even with finite slices" is true only in expectation, and the paper's own appendix contains a false claim about Gaussian projections.

What is genuinely new: the phi-weighting formulation is a tidy way to unify several SW variants, and Theorem 4.9 plus the ESSF interpretation gives a real explanation for why high-dimensional SWD underestimates distances. They are careful to disclaim the Gaussian-SW equivalence of Nadjahi et al., so the new piece is the projection change-of-variables argument. The experiments are fair: they sweep learning rates for all methods, use the variants' default hyperparameters, and report runtime and W2. That is credible evidence that a properly tuned classic SW is competitive.

Soft spots, in proportion: - The learning-rate absorption claim (Section 4.4) is asserted, not proven. Proposition 4.10 gives E[ESSF]=constant and convergence of the error as L grows, but for L=50, the classic MC SWD is not a scalar multiple of the subspace estimator. The stress-test note is correct: the product of two L-averages differs from the average over slices, and that difference interacts nonlinearly with the sorting inside each 1D Wasserstein term. In minibatch training the effective subspace also changes each step. So "informative slices for free" is a reasonable practical approximation, but the paper should either measure ESSF(L) variability during actual optimization or weaken the claim. - Appendix A.3, Proposition A.7 states that for any full-rank Gaussian, X/||X|| is uniform on the sphere. That is false unless the covariance is a scalar multiple of identity; the proof's diagonal step is also wrong. It is not load-bearing for Theorem 4.9, but it is an incorrect statement that needs correction. - No code or data artifacts are provided. Minor, but a paper whose message is "tune the learning rate and you are done" would benefit from releasing the sweep.

The central theorem holds up, and the empirical story is plausible and useful, but the practical claim is overstated. The paper deserves a serious referee; the right ask is to fix Prop A.7 and either add a direct test of ESSF variability or temper the Section 4.4 language. I would cite the theorem, and I would bring it to the group as a maybe—good counterweight to the "just sample informative slices" literature.

Referee Report

3 major / 6 minor

Summary. The paper studies the sliced-Wasserstein distance (SWD) for high-dimensional data whose support lies in a low-dimensional linear subspace. Under Assumption 4.1, it proves Theorem 4.9: the ambient SW_p^p is a fixed scalar multiple (C_k/C_d) of the SW_p^p computed in the effective subspace. It introduces a ϕ-weighting framework for rescaling 1D Wasserstein contributions by slice informativeness and argues that the per-slice reweighting simplifies to this single global factor, which can be absorbed into the learning rate in gradient-based learning. The paper reports experiments on gradient flow, color transfer, and deep generative modeling on FFHQ, showing that classical SWD with a well-chosen learning rate matches or outperforms more complex SW variants.

Significance. If the central claim holds, the paper provides a clean explanation of why the classical SWD can be competitive with data-adaptive SW variants: the ill-conditioning associated with uninformative slices is a global scaling phenomenon, rather than a per-slice one. The unifying ϕ-weighting formulation (Section 4.1) is a useful organizing framework. The population-level identity (Theorem 4.9) is exact and the synthetic verification (Section 5.1) is convincing. The paper also ships explicit convergence statements (Propositions 4.10 and 4.12) and an extensive experimental comparison with runtime tables. The main weakness is that the translation from the exact population identity to the finite-slice, minibatch setting used in all practical experiments is not rigorously established; this is the load-bearing step for the headline claim that informative slices come 'for free' via learning-rate search.

major comments (3)

[§4.4, Proposition 4.10] The claim that the Effective Subspace Scaling Factor (ESSF) 'can be absorbed into the learning rate' is only established in expectation and asymptotically. For the Monte Carlo estimator actually optimized in the experiments (L=50), the classic SWD is (1/L) Σ_l ||U^T θ_l||^p W_p^p(θ_k^l#µ_k, θ_k^l#ν_k), while the rescaled subspace estimator is [(1/L)Σ_l ||U^T θ_l||^p][(1/L)Σ_l W_p^p(θ_k^l#µ_k, θ_k^l#ν_k)]. These are equal only in expectation; their difference is nonzero for every finite L and enters the gradient nonlinearly through the sorting in each 1D Wasserstein term. Propositions 4.10(2) and 4.12 only give convergence to zero of this error as L→∞. With L=50, no finite-L bound is provided, and the paper reports no measurement of the per-iteration ESSF variability in the actual optimization tasks. The statement in Section 4.4 that the factor 'can be absorbed' is therefore an approximation, not a proven equivalence, and the learning-rate search in the experiments may be compensating for this mismatch. The authors should either provide finite-L gradient-error bounds or explicitly reframe the claim as a heuristic supported by the accompanying empirical study.
[§4.4, Remark 4.11, experiments] In minibatch training, the effective subspace U is not fixed across iterations: Remark 4.11 bounds its dimension by min{2B-1, d}, but the subspace itself changes with each minibatch. Consequently, the expected ESSF, C_k/C_d, is not a single constant throughout optimization; it drifts with U. The paper does not state any assumption under which this drift is negligible, and the experiments use a sweeping learning-rate grid that can absorb per-iteration scaling differences. To support the central claim, the authors should either derive a bound on the drift of the ESSF under minibatch updates or present empirical evidence (e.g., tracking ESSF values over training) that the drift is small relative to the learning-rate sensitivity.
[Appendix A.3, Proposition A.7] Proposition A.7 states that for any full-rank covariance Σ, f#N(0,Σ) equals the uniform distribution on S^{d-1}, where f(x)=x/||x||. This is false for anisotropic Gaussians: for Σ=diag(σ1^2,...,σd^2), the random direction X/||X|| has a density proportional to ∏ σ_i^{-1} exp(-x_i^2/(2σ_i^2)) on the sphere, which is not constant unless all σ_i are equal. The proof's Step 2 incorrectly identifies Σ^{-1/2}X/||Σ^{-1/2}X|| with X/||X||. The isotropic case used in Proposition A.9 is correct, so this error does not invalidate Theorem 4.9, but the misleading general statement and its proof must be corrected or removed.

minor comments (6)

[Appendix A.3, Remark A.8] Remark A.8 contains the typo 'distritbution' and the phrase 'rotationally invariant distritbution'; the sentence should read 'rotationally invariant distribution'.
[Section 5.2.3] The sentence 'The main observation is that all methods perform comparatively, aibeit different learning rates are required for good results' contains the typo 'aibeit' for 'albeit'.
[Theorem 4.9 vs. Appendix A.4] The appendix heading 'A.4 Relationship between the SWD in Rd and Rk' refers to 'Proposition 4.9' in the proof, but the result is numbered Theorem 4.9 in the main text; the cross-reference should be corrected.
[Appendix A.6.1] The subsection heading 'A.6.1 Proof of Theorem 4.10' appears before the proof of Proposition 4.10; the theorem number should be 'Proposition 4.10'.
[Figure 10 caption] Figure 10 is in the Color Transfer section but its caption reads 'Optimal basin plots for Gradient Flow with embedded synthetic datasets.' This caption should be updated to refer to the color transfer results.
[Proposition 4.10(3)] The concentration inequality in Proposition 4.10(3) is stated as P(ϵL < δ) ≥ 1 - e^{-δ^2L/K^2}, but the proof in Appendix A.6 writes a different expression; the constants and exponent should be checked for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the subspace scaling identity is derived from an external Gaussian-SW equivalence, and the learning-rate interpretation is an overclaim but not a circular reduction.

full rationale

Theorem 4.9's identity SW_p^p(µd,νd)=(C_k/C_d) SW_p^p(µk,νk) is proven by combining Proposition A.9—an external result of Nadjahi et al. (2021), which the paper explicitly disclaims as its own contribution—with the elementary projection identity in Lemma A.12/A.14. No parameter is fitted to data to produce the identity; the ratio C_k/C_d is a closed-form function of dimensions only. The ES-aligned informativeness φ_U(θ)=||U^Tθ|| and the reciprocal weight 1/||U^Tθ||^p are definitions, and Proposition 4.7's factorization is a mathematical fact rather than a renaming of the conclusion. The later claim that the global scale 'can be absorbed into the learning rate' is only asserted heuristically in Section 4.4: Proposition 4.10 and 4.12 give convergence in expectation and as L→∞, not exact proportionality for L=50, so the finite-L gradient mismatch is a correctness/rigor concern, not a circularity. The self-citations (Kolouri et al. 2019 for a standard concentration bound; Tran et al. 2024 in related work) are not load-bearing: the concentration inequality is an externally checkable mathematical fact. Thus no step reduces by construction to its own input.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on the explicit subspace assumption, on the prior Gaussian-SW equivalence of Nadjahi et al. (2021) (disclaimed in Remark A.10), and on standard probability facts. The learning-rate advice adds an unproven stability assumption about the scaling factor. No new physical entities are introduced.

free parameters (2)

Learning rate h = Grid-searched per task, e.g., {1,3,5,8} x 10^{-6} to 10^{2}
The central practical claim depends on selecting the optimal learning rate. The derived global scaling factor is not estimated or used directly; instead it is absorbed into this hyperparameter, which is fitted to the validation objective.
Number of random projections L = 50 for all experiments
The slice count is fixed by hand rather than derived. The variance of the ESSF estimator and the accuracy of the SW approximation depend on L, and the paper does not sweep it.

assumptions (5)

domain assumption Assumption 4.1: both measures are exactly supported on the same k-dimensional linear subspace V_k, with k minimal.
This is the structural premise of Proposition 4.7 and Theorem 4.9. Real data satisfies it only approximately, and the paper does not verify it for the image datasets.
standard math Gaussian-SW equivalence of Nadjahi et al. (2021): SW_p^p(µ,ν; N(0,I_d)) = C_d SW_p^p(µ,ν; U(S^{d-1})).
Used in the proof of Theorem 4.9 via Proposition A.9. The paper explicitly disclaims novelty for this statement in Remark A.10.
standard math For an isotropic Gaussian, the radius and direction are independent, and the normalized vector is uniform on the sphere.
Invoked through Proposition A.6(4) and the isotropic case of Proposition A.7 to derive the scaling constant and the uniform distribution of projected slice directions.
domain assumption Manifold hypothesis: high-dimensional image data is approximately supported on a low-dimensional manifold.
Motivates Assumption 4.1 for real datasets, citing Pope et al. (2021). It is not tested within the paper.
standard math For a minibatch of 2B samples, the empirical support lies in a subspace of dimension at most min{2B-1, d} after centering.
Remark 4.11 uses this to argue that the effective subspace assumption holds in minibatch learning, but it does not prove the scaling factor is constant across minibatches.

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Pith. "Pith review of Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices." pith.science (2026). https://pith.science/paper/ZQ7TCR3I

@misc{pith2026241110651,
  author       = {Pith},
  title        = {Pith review of: Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQ7TCR3I}},
  note         = {Machine review of arXiv:2411.10651}
}

read the original abstract

The practical applications of Wasserstein distances (WDs) are constrained by their sample and computational complexities. Sliced-Wasserstein distances (SWDs) provide a workaround by projecting distributions onto one-dimensional subspaces, leveraging the more efficient, closed-form WDs for one-dimensional distributions. However, in high dimensions, most random projections become uninformative due to the concentration of measure phenomenon. Although several SWD variants have been proposed to focus on \textit{informative} slices, they often introduce additional complexity, numerical instability, and compromise desirable theoretical (metric) properties of SWD. Amidst the growing literature that focuses on directly modifying the slicing distribution, which often face challenges, we revisit the classical Sliced-Wasserstein and propose instead to rescale the 1D Wasserstein to make all slices equally informative. Importantly, we show that with an appropriate data assumption and notion of \textit{slice informativeness}, rescaling for all individual slices simplifies to \textbf{a single global scaling factor} on the SWD. This, in turn, translates to the standard learning rate search for gradient-based learning in common machine learning workflows. We perform extensive experiments across various machine learning tasks showing that the classical SWD, when properly configured, can often match or surpass the performance of more complex variants. We then answer the following question: "Is Sliced-Wasserstein all you need for common learning tasks?"

Figures

Figures reproduced from arXiv: 2411.10651 by the authors.

**Figure 1.** Rescaling the 1D Wasserstein based on slice informativeness. The main challenge. Many machine learning problems involve high-dimensional data that has a low-dimensional structure. Formally, this phenomenon, known as the manifold hypothesis, states that for a dataset X ⊂ R d , there exists a k-dimensional manifold M where k ≪ d such that X approximately lies on M (Fefferman et al., 2016). For instance, rigorous dime… view at source ↗

**Figure 2.** Minibatches of d-dimensional data, with B source and B target samples, reside in a linear subspace with dimensionality at most k = min{2B − 1, d} when centered. Learning Rate Selection. While our results may initially suggest that the data must reside within a known subspace Vk of a specific dimension k, neither Vk nor k needs to be explicitly identified in practice. Moreover, Assumption 4.1 naturally holds in com… view at source ↗

**Figure 3.** Left: Illustration of two k-dimensional Gaussian distributions embedded in R d (500 samples each). Top row: Empirical ratios Cb with varying d for k = 2 and p = 1, 2. Bottom row: Empirical ratios Cb with varying k for in d = 1000 and p = 1, 2. Verifying Proposition 4.10 We proceed further to observe the empirical estimate ESSF \(L) = 1 L X L l=1 ∥U ⊤θ d l ∥ p and its variance for different values of L = {10, 50, 100… view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: The mean and standard deviation (shaded area) of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Classic synthetic 2D datasets (shown) embedded in spaces of different target dimensions. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Optimal basin plots for Gradient Flow with embedded synthetic datasets. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Optimal basin plots for MNIST(left) and CelebA(right). [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Gradient Flow visualization for images from the MNIST dataset (left) and the CelebA dataset (right). [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: 3 sets of source (top) and target (bottom) images for Color Transfer. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: Optimal basin plots for Gradient Flow with embedded synthetic datasets. [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: Left: Samples generated using different SW variants. Right: Optimal basin plot. 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Learning Rate 14 15 16 17 18 19 W 2 SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Learning Rate 15 16 17 1…

**Figure 12.** Figure 12: Optimal basin plots for M2F(left) and A2C(right). [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

**Figure 13.** Figure 13: Samples generated by the M2F (left) and (A2C) residual translators. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 14.** Figure 14: Set 1 (Color Transfer) Source Target SW (𝒲2 = 0.01) MaxSW (𝒲2 = 0.04) DSW (𝒲2 = 0.04) iMSW (𝒲2 = 0.03) oMSW (𝒲2 = 0.03) rMSW (𝒲2 = 0.03) viMSW (𝒲2 = 0.04) MaxKSW (𝒲2 = 0.04) EBSW (𝒲2 = 0.04) RPSW (𝒲2 = 0.11) EBRPSW (𝒲2 = 0.04) [PITH_FULL_IMAGE:figures/full_fig_p033_14.png]

**Figure 15.** Figure 15: Set 2 (Color Transfer) 33 [PITH_FULL_IMAGE:figures/full_fig_p033_15.png]

**Figure 16.** Figure 16: Set 3 (Color Transfer) (a) Left: Cumulative Explaning Variance plot for the FFHQ latents. Middle/Right: UMAP visualization of the Gender and Age splits. FFHQ Subset Train size Test size Adults (≥ 18) 48786 8104 Children (< 10) 8345 1405 Male 26732 4351 Female 32816 55…

**Figure 17.** Figure 17: The FFHQ dataset (Karras et al. (2019)) 43 [PITH_FULL_IMAGE:figures/full_fig_p043_17.png]

**Figure 18.** Figure 18: Visualization for the M2F translation task (using the model with the lowest [PITH_FULL_IMAGE:figures/full_fig_p047_18.png]

**Figure 19.** Figure 19: Visualization for the A2C translation task (using the model with the lowest [PITH_FULL_IMAGE:figures/full_fig_p048_19.png]

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Forward citations

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Adding SWGG dissimilarity constraints to sliced Wasserstein embedding, trained via primal-dual optimization with a softsort relaxation, improves pooling accuracy on image, point cloud, and protein-sequence benchmarks.

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Works this paper leans on

24 extracted references · 18 canonical work pages · cited by 1 Pith paper

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is a well-known result, and that is why isotropic Gaussian distribution is called a “rotationally invariant distritbution. ” We do not claim this proposition or its proof as contributions of this article; we present the proof merely for completeness. A.4 Relationship between the SWD in Rd and Rk In this section, we discuss the proof of the proposition 4.9...

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8345 1405 Male 26732 4351 Female 32816 5572 (b) Subset size. Figure 17: The FFHQ dataset (Karras et al. (2019)) 43 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 294.00 ± 0.38 287.00 ± 4.65 227.00 ± 4.40 275.00 ± 7.53286.00 ± 7.55 286.00 ± 7.36 294.00 ± 0.38201.00 ± 3.34 291.00 ± 0.71 291.00 ± 0.79291.00 ± 0.52 80 294.00 ± 0.38 286.00 ± ...

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35 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 1.8754±1.21833.6719±1.66692.2167±1.39101.1766±0.09012.0996±0.50582.3956±0.79933.0252±0.17062.6436±1.08823.0252±0.17062.6426±0.06692.8145±0.1491 80 1.6682±0.97271.8203±0.90882.2781±0.41021.1566±0.03182.0056±0.77641.8520±0.58242.2930±0.09031.8364±1.07022.2930±0.09032.0552±0.10702.3738±0.039...

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36 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.7900±0.07044.4368±0.09663.0902±0.10262.9073±0.39472.2985±0.12201.2282±0.05780.8969±0.05001.2429±0.116710.5129±0.22401.2812±0.016511.0450±0.8609 80 0.6047±0.08503.4799±0.05442.4792±0.26022.0292±0.27111.7091±0.09210.9662±0.02840.6897±0.04420.9003±0.03928.5668±0.39441.0411±0.05867.8665±0.8...

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38 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.7762±0.10234.1470±0.03522.9805±0.06422.5629±0.44432.0662±0.10621.2985±0.14300.7226±0.05451.0492±0.08378.9485±0.85841.2748±0.03328.9753±0.7660 80 0.6647±0.03593.2381±0.03672.2192±0.22202.1054±0.08411.6892±0.1...
[21]

39 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.8643±0.04802.4103±0.02433.0249±0.09922.8531±0.35981.9101±0.09040.9712±0.02270.8463±0.04780.9859±0.064516.7324±0.30580.8955±0.038016.3655±0.8700 80 0.6125±0.02641.7029±0.04442.2926±0.19292.2264±0.11021.5574±0...
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41 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.8453±0.00231.7094±0.03852.8643±0.23562.6594±0.12272.1708±0.07070.8819±0.04980.8580±0.01261.0229±0.081514.9925±0.91490.9446±0.052414.6610±0.9302 80 0.6462±0.01442.3625±1.63012.5657±0.09372.1892±0.19151.6570±0...
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37 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.7547±0.01595.2763±0.05812.7308±0.23563.2047±0.15532.2315±0.01521.1926±0.10090.7655±0.07251.0225±0.02598.2665±0.67631.2582±0.16258.2852±0.4012 80 0.7061±0.05884.4078±0.87412.1265±0.57072.6758±0.24321.5470±0.0...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

rotationally invariant distritbution

is a well-known result, and that is why isotropic Gaussian distribution is called a “rotationally invariant distritbution. ” We do not claim this proposition or its proof as contributions of this article; we present the proof merely for completeness. A.4 Relationship between the SWD in Rd and Rk In this section, we discuss the proof of the proposition 4.9...

work page

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34 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 2.7116±0.66160.6408±0.00793.2280±0.50941.3326±0.04112.2337±0.87272.5000±0.83351.3862±0.26032.5671±0.86202.8073±0.14162.6577±0.02822.7618±0.1329 80 2.1367±0.60870.5710±0.00411.0384±0.36401.3753±0.11061.7140±0.17031.8681±0.62681.5932±0.10412.5953±0.68612.3348±0.08282.2001±0.06312.2090±0.161...

work page arXiv 2001

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Nguyen, N

K. Nguyen, N. Ho, T. Pham, and H. Bui. Distributional sliced-wasserstein and applications to generative modeling. arXiv preprint arXiv:2002.07367,

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Figure 17: The FFHQ dataset (Karras et al

8345 1405 Male 26732 4351 Female 32816 5572 (b) Subset size. Figure 17: The FFHQ dataset (Karras et al. (2019)) 43 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 294.00 ± 0.38 287.00 ± 4.65 227.00 ± 4.40 275.00 ± 7.53286.00 ± 7.55 286.00 ± 7.36 294.00 ± 0.38201.00 ± 3.34 291.00 ± 0.71 291.00 ± 0.79291.00 ± 0.52 80 294.00 ± 0.38 286.00 ± ...

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It remains to show the convergence rate of error N . Since each ∥U ⊤θl∥p ∈ [0, 1], for each t > 0, by Hoeffding theorem, we have P(|Bn| ≥ϵ) ≤ e−2ϵ2L Replacing ϵ by ϵ/K, we have P(errorL ≤ ϵ) ≥ 1 − 2e 2ϵ2 L K2 and we complete the proof. A.6.1 Proof of Theorem 4.10 A.7 Special case: Learning rate bound for the SW Gradient Flow problem In this section, we co...

work page arXiv 2021

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35 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 1.8754±1.21833.6719±1.66692.2167±1.39101.1766±0.09012.0996±0.50582.3956±0.79933.0252±0.17062.6436±1.08823.0252±0.17062.6426±0.06692.8145±0.1491 80 1.6682±0.97271.8203±0.90882.2781±0.41021.1566±0.03182.0056±0.77641.8520±0.58242.2930±0.09031.8364±1.07022.2930±0.09032.0552±0.10702.3738±0.039...

work page arXiv

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36 LR SW MaxSW DSW MaxKSW iMSW viMSW oMSW rMSW EBSW RPSW EBRPSW 100 0.7900±0.07044.4368±0.09663.0902±0.10262.9073±0.39472.2985±0.12201.2282±0.05780.8969±0.05001.2429±0.116710.5129±0.22401.2812±0.016511.0450±0.8609 80 0.6047±0.08503.4799±0.05442.4792±0.26022.0292±0.27111.7091±0.09210.9662±0.02840.6897±0.04420.9003±0.03928.5668±0.39441.0411±0.05867.8665±0.8...

work page arXiv