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Coreset selection for the Sinkhorn divergence and generic smooth divergences

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For the Sinkhorn divergence, a convexly weighted coreset of $m = \omega(\log^d n)$ observed points reproduces the full-sample divergence up to an asymptotically negligible $o_p(1/n)$ error.

desk verdict A genuinely new polylog Sinkhorn coreset result with a coherent statistical core, but the advertised O(n^2m + m^3) construction time rests on an explicitly deferred Nyström proof that needs to be fixed before publication. read the letter →

arxiv 2504.20194 v2 pith:C7AYRD7U submitted 2025-04-28 stat.ML cs.LG

Alex Kokot , Alex Luedtke This is my paper · ORCID
classification stat.MLcs.LG
keywords coresetsSinkhorndivergencemaximummeandiscrepancyHadamarddifferentiabilityentropicoptimaltransportkernelquadraturerecombinationNyströmapproximation
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 CO2 (Coresets of Order 2), a two-step recipe for compressing a dataset with respect to any smooth divergence: expand the divergence to second order around the empirical distribution, which turns the task into a maximum mean discrepancy minimization, then compress against the kernel that appears in that expansion. The flagship application is the Sinkhorn divergence, the debiased entropically regularized optimal transport loss often used as a fast, smooth stand-in for Wasserstein distance. The paper claims that a convexly weighted coreset of $m = \omega(\log^d n)$ of the $n$ observed points reproduces the full-sample Sinkhorn divergence up to $S(P,P_m)=S(P,P_n)+o_p(1/n)$, matching random sampling at the $1/n$ scale with far fewer than $n$ points. If this is right, Sinkhorn coresets are the first provably asymptotically lossless $o(n)$-sized coresets for an optimal-transport-style loss, and coreset selection becomes a direct instance of kernel quadrature with a data-dependent kernel.

What carries the argument

The load-bearing object is the Hadamard operator $G$, the kernel of the divergence's second-order Taylor term at the population distribution: Lemma 1 shows that once $D$ is second-order Hadamard differentiable, $D(P_m)=D(P_n)+o_p(1/n)$ follows from $\mathrm{MMD}_G(P_n,P_m)=o_p(n^{-1/2})$, so compression reduces to MMD against $G$. For the Sinkhorn divergence the paper computes $G=\varepsilon(I-A^2)^{-1}\xi$, where $A$ is the entropic optimal transport operator $Af(x)=\int \xi(x,y)f(y)\,dP(y)$ and $\xi$ is the density of the entropic self-coupling, and proves this kernel is equivalent to the Gaussian kernel $\exp(-\|x-y\|^2/\varepsilon)$. The algorithm then builds the coreset by recombination on Nyström eigenvectors of the empirical $G_n$, and the geometric spectral decay of Gaussian Gram matrices supplies the $m=\omega(\log^d n)$ rate through Lemma 61.

What would settle it

Fix a compactly supported distribution in dimension $d$ and a regularization level $\varepsilon$; draw $n$ samples, build a CO2 coreset with $m=C\log^d n$ for several growing $n$, and compute $n|S(P,P_m)-S(P,P_n)|$ over repeated trials. Theorem 7 predicts this quantity converges to zero; observing bounded-away-from-zero values, or confidence intervals that do not shrink like $1/\sqrt{n}$, would refute the central guarantee.

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Extended reading notes

Core claim

The central discovery is that second-order Hadamard differentiability turns divergence compression into MMD compression: Lemma 1 shows that if $D$ is second-order Hadamard differentiable at $P$, then any $P_m$ with $\mathrm{MMD}_G(P_n,P_m)=o_p(n^{-1/2})$ satisfies $D(P_m)=D(P_n)+o_p(1/n)$, where $G$ is the kernel of the quadratic term. For the Sinkhorn divergence the paper verifies the previously unknown regularity needed to apply this: $S$ is second-order Hadamard differentiable relative to the Gaussian kernel RKHS, with Hadamard operator $G=\varepsilon(I-A^2)^{-1}\xi$ equivalent to the Gaussian kernel, and entropic optimal transport potentials are Hadamard differentiable in that refined topology. Because Gaussian Gram matrices have geometric spectral decay, the recombination-plus-Nyström MMD compressor needs only $m=\omega(\log^d n)$ points and $O(n^2m+m^3)$ time to reach the $o_p(1/n)$ target. The proof passes through new differentiability results for the exponentiated entropic potentials $(u_{\mu,\nu},v_{\mu,\nu})=(e^{-\phi_{\mu,\nu}/\varepsilon},e^{-\psi_{\mu,\nu}/\varepsilon})$.

Load-bearing premise

The load-bearing premise is that the Sinkhorn divergence's error as a function of the data distribution has a valid quadratic Taylor approximation in the very smooth function space induced by a Gaussian kernel, which the proof obtains from a technical local approximation and compact-support assumptions; if that Taylor approximation fails, the polylogarithmic guarantee collapses.

Editorial extensions

If this is right

  • With $m=\omega(\log^d n)$, Sinkhorn coresets achieve $S(P,P_m)=S(P,P_n)+o_p(1/n)$ in $O(n^2m+m^3)$ time, so asymptotic losslessness does not require a constant fraction of the data.
  • Since the reduction to MMD is generic, any improvement in convex MMD compression can be substituted into CO2 and immediately improves Sinkhorn coresets.
  • The new Hadamard differentiability results for entropic optimal transport potentials extend to other functionals built from entropic plans, such as the regularized OT cost itself or entropic barycenters, potentially giving them the same coreset guarantees.
  • In Sobolev-type RKHSs of order $\alpha$, the same framework gives $m=\omega(n^{1/(1-\beta)})$ points, so the required coreset size tracks the spectral decay of the kernel exactly.

Reading between the lines

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

  • The condition the paper actually needs is spectral equivalence between the divergence's second-order operator and a kernel with fast decay; entropically regularized Wasserstein losses with other ground costs are the natural testbed for whether the polylog result is specific to the Gaussian kernel or generic across smooth divergences.
  • The paper's reduction suggests a black-box implementation: estimate the second-order Hadamard operator by automatic differentiation of the divergence and compress against the resulting data-dependent kernel, avoiding the analytic derivation that currently limits CO2 to specially studied functionals.
  • The MNIST label-proportion result hints that Sinkhorn-optimal coresets are good for downstream aggregate statistics, not just for the divergence itself; a direct test would compare CO2 coresets against pure MMD coresets on classification and moment-estimation tasks.
  • The paper leaves the replacement of $Q_{G_n}$ by the cheaper $\varepsilon Q_{\xi_n}$ as a conjecture supported by Lemma 12; if that exponential-rate claim is correct, computing Sinkhorn coresets requires only one entropic plan and its spectrum.
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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

2 major / 4 minor

Summary. The paper introduces CO2, a two-stage framework for convexly weighted coreset selection with respect to generic smooth divergences. The first stage identifies the second-order Hadamard derivative of the divergence and the associated Hadamard operator, and the second stage performs MMD compression with respect to that operator using a Nyström-based recombination algorithm. The authors apply this framework to the Sinkhorn divergence, deriving new Hadamard differentiability results for entropic optimal transport potentials and showing that the Sinkhorn divergence is second-order Hadamard differentiable relative to a Gaussian RKHS. Their main theorem claims that, for m = ω(log^d n), a coreset of size m can be constructed in O(n^2 m + m^3) time such that S(P, P_m) = S(P, P_n) + o_p(1/n). Numerical experiments on Gaussian mixtures and MNIST illustrate the method's practical performance.

Significance. If the main result holds, this is a significant contribution: it gives the first polylogarithmic-size coreset guarantee for the Sinkhorn divergence, connects coreset selection to kernel quadrature through a functional Taylor expansion, and contains new regularity results for entropic optimal transport potentials that are of independent interest. The theoretical derivation is largely self-contained: the Sinkhorn kernel is derived from the divergence rather than fitted to data, the polylog bound follows from known geometric spectral decay of Gaussian kernels, and the proof of second-order Hadamard differentiability is detailed. The code is publicly available, and the experiments are clearly described. The principal weakness is that the advertised fast construction time is not fully supported, because the key Nyström transfer argument in Lemma 12 is explicitly deferred rather than proved.

major comments (2)
  1. [§4.3, Lemma 12; Theorem 7] Theorem 7's O(n^2 m + m^3) construction time is not fully supported, because the proof of Lemma 12 only treats the exact rank-r SVD variant and explicitly defers the Nyström transfer: 'We leave a complete argument to future research, as the necessary Nyström manipulations are outside the scope of this paper.' The proof assumes that the residual vector w satisfies a_1 = ... = a_m = 0 in the eigenbasis of π_{P_n,P_n}, which is a property of exact diagonalization and subsequent exact recombination, not of the Nyström approximation used in Algorithm 3. Since Algorithm 3 and Section 5 operate with the Nyström approximation of G or of ξ_n, the stated polylogarithmic guarantee at O(n^2 m + m^3) time remains conditional. A complete proof of the Nyström transfer, or a revised Theorem 7 that states the O(n^3) exact-SVD construction, is needed.
  2. [§4.3, Lemma 12; §5] There is an ambiguity about which quadratic form the implemented algorithm minimizes: Section 5 says 'CO2 is performed with respect to π_{P_n,P_n} for improved computational efficiency as detailed in Section 4.3,' while Algorithm 3 calls Nyström(G) on the Hadamard operator G. Lemma 12 only relates Q_{G_n} to εQ_{ξ_n} in the exact-SVD case. The paper does not provide a theorem showing that the Nyström-approximated directions of π_{P_n,P_n} yield the same or sufficiently close guarantee as exact SVD, so the empirical procedure and the mathematical claim in Theorem 7 are not yet matched.
minor comments (4)
  1. [§3.1] The word 'ambguity' should be 'ambiguity'.
  2. [Theorem 7] Theorem 7 states O(n^2 m + m^3), but Lemma 2 and Theorem 3 include θ^3 m^3 and m^3 log(n/m) factors; the theorem should either state the assumptions on the oversampling parameter θ or present the complexity consistently with the lemmas it relies on.
  3. [Algorithm 2 and Algorithm 5] Algorithm 2 defines V as the orthogonal complement of U⊕1, whereas Algorithm 5 uses U⊕1⊕kdiag; the two algorithms should use the same complement, or the difference should be justified in the text.
  4. [Proof of Lemma 12] The claim that the eigenvalues λ_i of π_{P_n,P_n} are geometrically decaying by 'Lemmas 22 and 61' is not self-evident; the argument should explicitly explain how uniform smoothness of the Sinkhorn scalings and the Gaussian RKHS spectral decay combine to yield this geometric decay for the empirical Sinkhorn plan.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Sinkhorn coreset guarantee is derived from Hadamard differentiability and external spectral decay results, not from fitting or self-citation.

full rationale

Walking the derivation chain, I find no step in which a claimed prediction or guarantee is identical, by construction, to a fitted input or to a prior self-citation. The central statistical claim (Theorem 7) is assembled from (i) Lemma 1, a general second-order delta-method bound that transfers MMD control to any second-order Hadamard differentiable divergence; (ii) Theorem 9, which computes the Sinkhorn Hadamard operator G = ε(I−A²)⁻¹ξ by differentiating the divergence rather than imposing it; and (iii) Lemma 61, which supplies the Gaussian kernel’s geometric spectral decay from external results (Ma and Belkin; Yang et al.). The Sinkhorn regularity results (Theorems 8 and 48) are proved via an implicit-function-theorem argument and use external references (Goldfeld et al.; Gonzalez-Sanz et al.) for the Schrödinger system and local expansions, not as self-supporting citations. The only load-bearing step with an explicit caveat is computational Lemma 12, which states 'We leave a complete argument to future research' for the Nyström transfer and proves only the exact-SVD variant; Theorem 7’s time-complexity claim is therefore not fully supported. That is an incompleteness or correctness gap, not a circular reduction, because the statistical guarantee does not presuppose the conclusion it derives. The sole self-citation (Luedtke 2024) appears in the discussion as a pointer to automatic differentiation and is not load-bearing. Accordingly the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard RKHS and delta-method theory, plus domain assumptions about compact support and eigenfunction boundedness. No new postulated entities (particles, forces, or conserved quantities) are introduced.

free parameters (1)
  • Sinkhorn regularization epsilon = 2d (synthetic), 1.5e4 (MNIST)
    User-chosen regularization parameter in the Sinkhorn divergence. The theoretical result holds for any fixed epsilon, but the experiments select it by heuristic, and it controls the Gaussian kernel bandwidth and spectral decay, influencing the required polylog order.
assumptions (6)
  • domain assumption The data distribution P is compactly supported on X subset R^d.
    Stated in Section 1; needed for compactness, uniform bounds, and spectral decay arguments throughout the paper.
  • domain assumption RKHS eigenfunctions are uniformly bounded in sup-norm, ||psi_i||_infty <= M.
    Assumed in Section 2.1 to relate Gram matrix spectral decay to the population Mercer decomposition; known for Matern and Gaussian kernels.
  • domain assumption The divergence D is second-order Hadamard differentiable at P relative to the RKHS topology.
    Core condition for Lemma 1 and Theorem 3; verified for the Sinkhorn divergence in Theorem 48 using the Gaussian kernel RKHS.
  • standard math The Gaussian kernel on a compact domain has geometrically decaying eigenvalues with sup-norm bounded eigenfunctions.
    Used in Lemma 61 to obtain m = omega(log^d n); cited from Yang et al. 2020 and Ma and Belkin 2017.
  • domain assumption Entropic optimal transport potentials satisfy the Schrodinger system and can be uniquely normalized.
    Used in Section 4.2 and Appendix E.1, following Csiszar 1975 and Ruschendorf and Thomsen 1993; necessary to define u and v and to apply the implicit function theorem.
  • domain assumption Perturbations in the Hadamard differentiability analysis are uniformly supported on a compact set X.
    Assumed before Lemma 45 to ensure uniform convergence and integrability of potential derivatives.

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Cite this review

Pith. "Pith review of Coreset selection for the Sinkhorn divergence and generic smooth divergences." pith.science (2026). https://pith.science/paper/C7AYRD7U

@misc{pith2026250420194,
  author       = {Pith},
  title        = {Pith review of: Coreset selection for the Sinkhorn divergence and generic smooth divergences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C7AYRD7U}},
  note         = {Machine review of arXiv:2504.20194}
}
read the original abstract

We introduce CO2, an efficient algorithm to produce convexly-weighted coresets with respect to generic smooth divergences. By employing a functional Taylor expansion, we show a local equivalence between sufficiently regular losses and their second order approximations, reducing the coreset selection problem to maximum mean discrepancy minimization. We apply CO2 to the Sinkhorn divergence, providing a novel sampling procedure that requires poly-logarithmically many data points to match the approximation guarantees of random sampling. To show this, we additionally verify several new regularity properties for entropically regularized optimal transport of independent interest. Our approach leads to a new perspective linking coreset selection and kernel quadrature to classical statistical methods such as moment and score matching. We showcase this method with a practical application of subsampling image data, and highlight key directions to explore for improved algorithmic efficiency and theoretical guarantees.

Figures

Figures reproduced from arXiv: 2504.20194 by the authors.

Figure 1
Figure 1. Eight repetitions of the simulation described in Section 5.1. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. The reconstruction error Sε(Pn, Pm) in various dimensions (left) and dataset sizes n (right). In the first plot the sample size is fixed at n = 2.5 × 104 , for the latter the dimension is fixed at d = 10. distribution in dimensions d = 2, 5, 10, and we set the regularization parameter to be ε = 2d in each setting. We then compare Sε(Pn, Pm) for Pm generated by Sinkhorn compression to Pm generated by random sampling.… view at source ↗
Figure 3
Figure 3. Percent relative error of S¨(Pn, Pm) compared to S(Pn, Pm) with Pm generated via recombination compression. algorithm of their preference at this step of our method. This is particularly useful as this is a growing area of research, and we expect the development of algorithms that greatly improve our basic recombination approach, particularly in terms of practical finite sample performance. We draw n = 2.5 × 104 sam… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: MMD compression algorithms as well as other sampling methods employed at the secondary stage of our Sinkhorn compression method and the resulting compres￾sion error S(Pn, Pm) induced by these algorithms. enforcing that our coreset belongs to the observed dataset, there…
Figure 5
Figure 5. Figure 5: PCA coordinates of MNIST data with a Sinkhorn CO2 coreset overlaid on the sample, with figure sizes proportional to the sample weights. For interpretability, only digits 1, 2, and 3 are displayed here. 6 Discussion The CO2 algorithm attains rapid, asymptotically lossle…
Figure 6
Figure 6. Figure 6: 1000 MNIST digits selected via Sinkhorn CO2, sorted by digit and with areas proportional to their weights. Note: the treemap software used to produce this visualization could not maintain the 1:1 aspect ratio of the original MNIST images [PITH_FULL_IMAGE:figures/full_…
Figure 7
Figure 7. Figure 7: Q-Q plots of the Sinkhorn reconstruction error (left) and ℓ1 error between the label proportions (right) of the compressed data as compared to random samples. asymptotically minimax compression for all sufficiently regular divergences. Likewise, we speculate that in fi…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.