Pith. sign in

REVIEW 5 minor 34 references

Nearly sharp comparison results for sliced and max-sliced Wasserstein distances

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves the Hölder exponent 2/(d+2) for max-sliced Wasserstein distance on the unit ball is optimal in every dimension, and that structural bounds need a nearly linear complexity factor.

desk verdict A clean, citable resolution of the optimal exponent for max-sliced Wasserstein comparison, with a genuinely new construction; the only real soft spot is the unproved import of a discrepancy bound for the lower-bound half. read the letter →

arxiv 2608.13374 v1 pith:QGUBKZX5 submitted 2026-08-13 math.PR math.FAmath.STstat.TH

Jonathan Niles-Weed , Jacob Shkrob This is my paper · ORCID
classification math.PRmath.FAmath.STstat.TH MSC 60B1049Q2260D0511K38
keywords slicedWassersteindistancemax-slicedHöldercomparisonexponentsoptimaltransporthalf-spacediscrepancyVoronoimatchingKantorovich–Rubinsteinnormrandomprojectionsanticoncentration
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 pins down how much of the Wasserstein distance $W_1$ between two distributions survives when one records only one-dimensional projections, either averaged (sliced) or worst-case (max-sliced). It proves that the known upper bound $W_1 \le C \widetilde{W}_{1,1}^{2/(d+2)}$ on the unit ball cannot be improved: for every $d\ge 2$ and small $\rho$ there are measures on the ball with $W_1 \ge \rho$ but $\widetilde{W}_{1,1} \le C_d \rho^{1+d/2}$, so no comparison with a larger Hölder exponent exists. Under a nearest-atom matching condition, it proves $W_p \le C \sqrt{d}\, K \,\mathrm{SW}_{p,1}$, with an analogous bound for $k$-dimensional projections replacing $K$ by $K^{(k)} \le N^{1/k}$. It also proves, via a half-space discrepancy construction, that the linear factor $K$ is necessary up to a polylogarithmic factor.

What carries the argument

The central identity is Condition 1.2, under which $W_p^p(\mu,\nu)$ equals $\int d(x,Y)^p\, d\mu(x)$ (Lemma 3.1), because the optimal coupling is exactly the nearest-atom Voronoi matching. The proof of the upper bound then reduces to an anticoncentration estimate for uniform random projections: for $U$ uniform on the Stiefel manifold $G_{d,k}$, $\mathbb{P}(\|Uv\| < t\sqrt{k/d}\,\|v\|) \le (C_0 t)^k$ (Lemma 3.3), which turns a small average projected distance into a pointwise lower bound on $\mathbb{E}[d(Ux,UY)^p]$ in terms of $d(x,Y)/K^{(k)}$. For the sharpness of the Hölder exponent, the machinery is a 'star of Gaussian pancakes': signed one-dimensional perturbations whose moments up to order $d+1$ vanish, planted along an $\epsilon$-packing of directions in $\mathbb{RP}^{d-1}$, with Lemma 2.2 bounding the projected Kantorovich–Rubinstein norm by $C r \min(\epsilon, \epsilon^{d+2}/\beta^{d+1})$ and the local entropy bound (2.16) limiting how many pancakes any fixed direction sees. The lower bound on $K$ uses the cited half-space discrepancy estimate, which says the lattice of cube centers has $L^1$ average half-space discrepancy $O((\log N)^d)$, giving $\mathrm{SW}_{1,1} \le C_d (\log N)^d / N$.

What would settle it

Evaluate numerically, for $d=3$ and increasing $M$, the quantity $\sup_t \int_{S^{d-1}} |\mathrm{card}(\Lambda_M \cap P_{\theta,t}) - N 2^{-d}|P_{\theta,t}||\, d\sigma(\theta)$ for the $N=(2M+1)^d$ lattice in $[-1,1]^d$; if this discrepancy grows faster than $(\log N)^d$, then the bound in Lemma 4.1(iv) collapses and the proof of Theorem 1.4 fails. Alternatively, produce a pair $(\mu,\nu)$ satisfying Condition 1.2 with $K_{\mu,\nu}=K$ and $W_1(\mu,\nu)/\mathrm{SW}_{1,1}(\mu,\nu) = o(K/(\log K)^d)$, which would refute the claimed near-optimality directly.

Watch

Extended reading notes

Core claim

On the paper's own terms, the headline findings are two. First (Theorem 1.1), the max-sliced 1-Wasserstein distance cannot control $W_1$ on $\mathcal{P}(B_1)$ with a Hölder exponent better than $2/(d+2)$: the authors construct explicit pairs $(\mu_\epsilon, \nu_\epsilon)$ with $W_1(\mu_\epsilon, \nu_\epsilon) \asymp \epsilon^{2-2/d}$ and $\widetilde{W}_{1,1}(\mu_\epsilon, \nu_\epsilon) \lesssim \epsilon^{(d+1)-2/d}$, and the ratio of exponents is exactly $1+d/2$, forcing $\beta \le 2/(d+2)$ when combined with the known upper bound. Second (Theorems 1.3 and 1.4), under Condition 1.2 — that some optimal coupling matches every $\mu$-mass point to a nearest atom of the discrete measure $\nu$ — the comparison $W_p \le C \sqrt{d}\, K_{\mu,\nu} \,\mathrm{SW}_{p,1}$ holds with $K_{\mu,\nu}$ the essential supremum of $d(x,Y) \sum_{y\in Y} 1/\|x-y\|$, and the linear dependence on $K$ is near-optimal: there are examples with $K_{\mu,\nu} \asymp_d K$ and $W_1 \ge c_d K/(\log(K+1))^d \,\mathrm{SW}_{1,1}$. The $k$-dimensional projection version (Theorem 1.5) replaces $K$ by $K^{(k)} \le N^{1/k}$ and the factor $\sqrt{d}$ by $\sqrt{d/k}$.

Load-bearing premise

The load-bearing external input is the half-space discrepancy bound for the cubic lattice imported from the cited reference and stated as (4.2); if the true discrepancy of this lattice is larger than polylogarithmic, the lower bound $W_1/\mathrm{SW}_{1,1} \gtrsim K/(\log K)^d$ no longer follows, and the near-optimality of the $K$ factor in Theorem 1.3 is not established.

Editorial extensions

If this is right

  • For $p=1$ on the unit ball, the comparison $W_1 \le C \widetilde{W}_1^\beta$ cannot hold with $\beta > 2/(d+2)$, so the max-sliced distance is exponentially coarser than $W_1$ in high dimension.
  • If $\nu$ is discrete and the optimal transport is a nearest-atom matching, then $W_p$ and $\mathrm{SW}_{p,1}$ become equivalent up to the factor $\sqrt{d}\,K$, so the information loss of slicing is governed by the local crowding $K$ rather than by the total number of atoms.
  • Using $k$-dimensional projections improves the constant: for $N$ atoms the multiplier is at most $C\sqrt{d/k}\, N^{1/k}$, so higher-dimensional projections are strictly less wasteful.
  • The near-linear dependence on $K$ is intrinsic: some pairs satisfying Condition 1.2 force $W_1 \ge c_d K/(\log(K+1))^d \,\mathrm{SW}_{1,1}$, so no method can remove the $K$ factor entirely below polylog.
  • Since $\mathrm{SW}_{p,1} \le \widetilde{W}_{p,1}$, the same bounds and lower-bound construction hold verbatim with the max-sliced distance in place of the sliced distance.

Reading between the lines

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

  • A practical reading is that algorithms which replace Wasserstein by sliced Wasserstein on discrete data should expect an effective cost factor comparable to $K$, i.e., to the number of atoms within a few multiples of the nearest-neighbour distance; this is testable on point clouds by computing $K$ empirically and comparing $W_1/\mathrm{SW}_{1,1}$.
  • The discrepancy input of Theorem 1.4 is the only non-elementary step; substituting any point set with sub-polylogarithmic $L^1$ half-space discrepancy into the same cube construction would yield the same lower bound, suggesting the $K/(\log K)^d$ phenomenon is a projection-averaging effect rather than a special feature of the lattice.
  • The moment-matching 'pancake' construction is tailored to $p=1$ through the Kantorovich–Rubinstein norm; a natural extension is to build analogous signed perturbations with more vanishing moments to test whether the optimal exponents for $p>1$ obey the same dimensional barrier.
  • For Gaussian mixtures with well-separated components, $K$ should be close to $1$, so Theorem 1.3 predicts near-Lipschitz $W_p$--$\mathrm{SW}_{p,1}$ equivalence in that regime — a quantitative prediction that can be checked by simulation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. This paper studies quantitative comparisons between the p-Wasserstein distance W_p and its sliced and max-sliced analogues SW_{p,k} and tilde-W_{p,1} on Euclidean space. Theorem 1.1 constructs, for every d≥2 and every sufficiently small rho>0, probability measures μ,ν supported on the unit ball such that W_1(μ,ν)≥rho and tilde-W_{1,1}(μ,ν)≤C_d rho^{1+d/2}; this proves that the Hölder exponent 2/(d+2) obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance is optimal for every dimension. Theorems 1.3 and 1.5 establish that under Condition 1.2 (an optimal coupling transporting μ-a.e. point to a nearest atom of a discrete measure ν), W_p(μ,ν)≤C√(d/k) K^{(k)}_{μ,ν} SW_{p,k}(μ,ν), where the complexity parameter K^{(k)} is at most the number of atoms and can be substantially smaller. Theorem 1.4, based on the Chen–Travaglini half-space discrepancy estimate, shows that the linear dependence on K in Theorem 1.3 cannot be improved by more than a polylogarithmic factor. The paper also contains a useful comparison with the recent bound of Park and Slepčev and a discussion of the optimal-exponent results of Carlier, Figalli, Mérigot, and Wang.

Significance. These results are significant. The sharpness of the exponent 2/(d+2) for max-sliced Wasserstein distances resolves a question left open in Bobkov and Götze's work, and the construction is a nontrivial adaptation of the Gaussian-pancake method to the averaging regime. The structural comparison under Condition 1.2 is clean and likely useful: it replaces a smallness assumption on W_infty in the Park–Slepčev theorem by a transport-geometric condition and gives explicit dependence on the complexity parameter. The proof of the upper bounds is self-contained, with a careful anticoncentration lemma on the Stiefel manifold, and the calculations in Section 2 are checkable; I verified the exponent bookkeeping and the packing/sparsity constraints. The lower bound in Theorem 1.4 is less self-contained because it imports the Chen–Travaglini discrepancy bound, but this is a legitimate external input and the way it is converted into the sliced-Wasserstein lower bound is correct. Overall the manuscript is a strong contribution to the optimal-transport literature.

minor comments (5)
  1. [Section 4, Eq. (4.2)] The entire quantitative content of Theorem 1.4 is inherited from the Chen–Travaglini discrepancy bound. As written, (4.2) involves sup_t outside the angular integral. If the cited result is the usual integrated L1 form with ∫∫ |...| dt dσ, please state that form explicitly and rewrite Lemma 4.1(iv) accordingly; this is a local fix but important for the reader to verify the constant's dependence.
  2. [Theorems 1.3 and 1.5] The comparison is non-vacuous only when K^{(k)}_{μ,ν}<∞; the statements should say this explicitly. The proof already says 'we may assume K<∞', but the theorem statements should not leave the reader to infer that the sum in (1.15) is finite.
  3. [Eq. (2.22)] There is an index typo: in 1{d(x+sρu_j, P_i)≤|s|ρ} the vector should be u_i, since the point belongs to the i-th slab; as written it conflicts with the outer summation index j.
  4. [Lemma 2.2 proof] After reducing to z=0, the shorthand λ^{(ρ)}_{(u,r)} and λ_{(u,r)} is introduced without an explicit definition; please add a sentence identifying these as λ^{(ρ)}_{(u,0,r)} and λ_{(u,0,r)}.
  5. [Typesetting and wording] There are formatting artifacts in the typeset text, e.g., 'itsufficestobound' in Section 2.3 and missing spaces in the display after (2.13); please correct. Also, in Section 4, 'By the symmetry, the bound (4.2) holds for all t∈R' would be clearer if it said 'by reflecting θ to -θ and t to -t'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected: the headline theorems are proved from explicit constructions, self-contained lemmas, and independent external citations; the sole self-citation is a non-load-bearing remark.

full rationale

The derivation chain is self-contained in all three main results. Theorem 1.1 is proven by an explicit construction in Section 2: the signed measure sigma_epsilon is assembled from the moment-matched weight g and shifted disks, Lemmas 2.2-2.5 are proved in the paper, and the lower and upper bounds follow by Kantorovich duality and a dyadic summation over the epsilon-packing. The exponent 2/(d+2) emerges from the chosen scalings r = a epsilon^{(d-2)/d}, rho = r epsilon, and m ~ epsilon^{-(d-1)}; it is not inserted from the target bound. Theorems 1.3 and 1.5 are proved from Condition 1.2, the closed-form identity in Lemma 3.1, the projection lower bound in Lemma 3.2, and the self-contained anticoncentration estimate Lemma 3.3; the constant K appears through its definition (1.11)/(1.15), not as a fitted quantity, and no quantity is renamed as a prediction. Theorem 1.4 uses the Chen-Travaglini half-space discrepancy bound (4.2) as an external imported estimate; that import is a correctness risk if the cited theorem were misstated, but it is not circular because the construction and the conversion to sliced Wasserstein distance are independent of the claimed conclusion. The only self-citation, [CNR25], appears in a remark about the known curse of dimensionality of empirical Wasserstein convergence and is not load-bearing for any theorem. No step reduces by construction to its own inputs, so the paper receives circularity score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is largely self-contained. The main imported results are standard (Kantorovich duality, 1D CDF representation, projective-space entropy bounds, Beta anticoncentration) plus the Chen-Travaglini discrepancy theorem. No parameters are fitted to data; constants are universal. The complexity parameter K is a definition, not an invented entity.

assumptions (6)
  • standard math Kantorovich-Rubinstein duality: W1(µ,ν) = ||µ−ν||_KR for equal total mass.
    Used throughout Section 2 to translate Wasserstein bounds into KR norm estimates; see [Vil03, Theorem 1.14].
  • standard math L1 representation of 1D Wasserstein distance as the L1 distance between cumulative distribution functions.
    Used in Lemma 2.2 and Lemma 4.1(iv); see [BL19].
  • standard math Local entropy bounds for ε-packings of RP^{d-1} with the projective metric (Lemma 2.3).
    Imported from [Paj98, Sza82, PV14]; the packing size and the local counting bound (2.16) drive the max-sliced upper bound.
  • standard math A random projection squared norm has Beta(k/2,(d-k)/2) distribution.
    Used in the proof of Lemma 3.3; see [FM90].
  • standard math Chen-Travaglini L1 half-space discrepancy bound (4.2) for the lattice Λ_M.
    External theorem imported for Theorem 1.4; not proved in the paper. The bound is used in Lemma 4.1(iv) to control SW_{1,1}.
  • standard math Existence of a smooth signed weight g on (-1,1) with ∫g=1, ∫s^j g=0 for j=1..d+1, and ∫|s|g>0.
    Stated in Section 2.2; follows from linear independence of the functions involved. Essential for moment-matched perturbations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nearly sharp comparison results for sliced and max-sliced Wasserstein distances." pith.science (2026). https://pith.science/paper/QGUBKZX5

@misc{pith2026260813374,
  author       = {Pith},
  title        = {Pith review of: Nearly sharp comparison results for sliced and max-sliced Wasserstein distances},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGUBKZX5}},
  note         = {Machine review of arXiv:2608.13374}
}
abstract

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the H\"older exponent~$\frac{2}{d+2}$ obtained by Bobkov and G\"otze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slep\v{c}ev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

Figures

Figures reproduced from arXiv: 2608.13374 by the authors.

Figure 1
Figure 1. The star of Gaussian pancakes in the illustrative case d = 2: the mixtures µ (left) and ν (right) and their one-dimensional projections θ#µ and θ#ν along a direction θ. (For legibility, the perturbation depicted is a symmetric two-point shift of each pancake, which matches only the first moment.) Standard Kantorovich duality [Vil03, Theorem 1.14] shows that W1(µ, ν) = ∥µ − ν∥KR. We therefore use the latter quantity … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 28 canonical work pages

  1. [1]

    Chen, William W. L. and Travaglini, Giancarlo , journal=. An. 2011 , doi=

    work page 2011

  2. [2]

    Geometry and analytic properties of the sliced

    Park, Sangmin and Slep. Geometry and analytic properties of the sliced. Journal of Functional Analysis , volume=. 2025 , publisher=

    work page 2025

  3. [3]

    Generalized sliced

    Kolouri, Soheil and Nadjahi, Kimia and Simsekli, Umut and Badeau, Roland and Rohde, Gustavo , booktitle=. Generalized sliced

    work page

  4. [4]

    , journal=

    Boedihardjo, March T. , journal=. Sharp bounds for max-sliced. 2026 , doi=

    work page 2026

  5. [5]

    One-Dimensional Empirical Measures, Order Statistics, and

    Bobkov, Sergey and Ledoux, Michel , series=. One-Dimensional Empirical Measures, Order Statistics, and. 2019 , publisher=

    work page 2019

  6. [6]

    Indiana University Mathematics Journal , volume=

    Sliced optimal transport: is it a suitable replacement? , author=. Indiana University Mathematics Journal , volume=

    work page

  7. [7]

    Continuous

    Bruna, Joan and Regev, Oded and Song, Min Jae and Tang, Yi , booktitle=. Continuous

    work page

  8. [8]

    International Conference on Scale Space and Variational Methods in Computer Vision (SSVM) , pages=

    Rabin, Julien and Peyr. International Conference on Scale Space and Variational Methods in Computer Vision (SSVM) , pages=. 2011 , organization=

    work page 2011

Show all 34 references
  1. [9]

    2013 , school=

    Unidimensional and evolution methods for optimal transportation , author=. 2013 , school=

    2013

  2. [10]

    Statistical Optimal Transport:

    Chewi, Sinho and Niles-Weed, Jonathan and Rigollet, Philippe , series=. Statistical Optimal Transport:. 2025 , doi=

    2025

  3. [11]

    Liutkus, Antoine and Simsekli, Umut and Majewski, Szymon and Durmus, Alain and St. Sliced-. International Conference on Machine Learning , pages=. 2019 , organization=

    2019

  4. [12]

    2024 , publisher=

    Rousseau, Judith and Scricciolo, Catia , journal=. 2024 , publisher=

    2024

  5. [13]

    , booktitle=

    Deshpande, Ishan and Hu, Yuan-Ting and Sun, Ruoyu and Pyrros, Ayis and Siddiqui, Nasir and Koyejo, Sanmi and Zhao, Zhizhen and Forsyth, David and Schwing, Alexander G. , booktitle=. Max-sliced

  6. [14]

    International Conference on Machine Learning , pages=

    Adversarial examples from computational constraints , author=. International Conference on Machine Learning , pages=. 2019 , organization=

    2019

  7. [15]

    Efficient gradient flows in sliced-

    Bonet, Cl. Efficient gradient flows in sliced-. Transactions on Machine Learning Research , year=

  8. [16]

    and Stewart, Alistair , booktitle=

    Diakonikolas, Ilias and Kane, Daniel M. and Stewart, Alistair , booktitle=. Statistical query lower bounds for robust estimation of high-dimensional. 2017 , organization=

    2017

  9. [17]

    Annals of the Institute of Statistical Mathematics , volume=

    Some geometric applications of the beta distribution , author=. Annals of the Institute of Statistical Mathematics , volume=. 1990 , publisher=

    1990

  10. [18]

    Minimax confidence intervals for the sliced

    Manole, Tudor and Balakrishnan, Sivaraman and Wasserman, Larry , journal=. Minimax confidence intervals for the sliced. 2022 , publisher=

    2022

  11. [19]

    and Quinto, Eric Todd , journal=

    Hahn, Marjorie G. and Quinto, Eric Todd , journal=. Distances between measures from 1-dimensional projections as implied by continuity of the inverse. 1985 , publisher=

    1985

  12. [20]

    Stability of the

    Terzioglu, Fatma and Murray, Ryan , journal=. Stability of the

  13. [21]

    Sharp comparisons between sliced and standard

    Carlier, Guillaume and Figalli, Alessio and M. Sharp comparisons between sliced and standard. arXiv preprint arXiv:2510.16465 , year=

  14. [22]

    Bobkov, Sergey G. and G. Quantified. arXiv preprint arXiv:2412.10276 , note=

  15. [23]

    Journal of Functional Analysis , volume=

    Neighborhoods on the Grassmannian of marginals with bounded isotropic constant , author=. Journal of Functional Analysis , volume=. 2014 , publisher=

    2014

  16. [24]

    Metric entropy of the

    Pajor, Alain , booktitle=. Metric entropy of the. 1998 , publisher=

    1998

  17. [25]

    Strong equivalence between metrics of

    Bayraktar, Erhan and Guo, Gaoyue , journal=. Strong equivalence between metrics of. 2021 , doi=

    2021

  18. [26]

    Erratum to:

    Bayraktar, Erhan and Guo, Gaoyue , journal=. Erratum to:. 2025 , doi=

    2025

  19. [27]

    Champion, Thierry and De Pascale, Luigi and Juutinen, Petri , journal=. The. 2008 , doi=

    2008

  20. [28]

    Sliced and

    Bonneel, Nicolas and Rabin, Julien and Peyr. Sliced and. Journal of Mathematical Imaging and Vision , volume=. 2015 , doi=

    2015

  21. [29]

    Optimal estimation of

    Wu, Yihong and Yang, Pengkun , journal=. Optimal estimation of. 2020 , doi=

    2020

  22. [30]

    2003 , publisher=

    Topics in Optimal Transportation , author=. 2003 , publisher=

    2003

  23. [31]

    Szarek, Stanis. Nets of. Proceedings of Banach Spaces Workshop , publisher=

  24. [32]

    arXiv preprint math/9701213 , year=

    Metric entropy of homogeneous spaces , author=. arXiv preprint math/9701213 , year=

    arXiv

  25. [33]

    International conference on machine learning , pages=

    Subspace robust Wasserstein distances , author=. International conference on machine learning , pages=. 2019 , organization=

    2019

  26. [34]

    , journal=

    Doss, Natalie and Wu, Yihong and Yang, Pengkun and Zhou, Harrison H. , journal=. Optimal estimation of high-dimensional. 2023 , doi=

    2023

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.