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REVIEW 2 major objections 6 minor 145 references

Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD

T0 review · 2 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Estimating MMD, HSIC and KSD cannot beat the parametric n^{-1/2} rate on general spaces under mild kernel assumptions.

desk verdict Clean Le Cam lower bounds that close the topological-space gap for MMD/HSIC/KSD, with one overstated tightness claim for unbounded kernels. read the letter →

arxiv 2607.24235 v1 pith:2WBBHXPP submitted 2026-07-27 stat.ML cs.LGmath.STstat.TH

Jose Cribeiro-Ramallo , Florian Kalinke , Zoltán Szabó This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH
keywords maximummeandiscrepancyHilbert-SchmidtindependencecriterionkernelSteinminimaxlowerboundreproducingHilbertspaceembeddingperturbationsLeCammethod
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

Kernel discrepancies such as MMD, HSIC and KSD are standard tools for comparing distributions, testing independence and checking goodness of fit. Existing estimators already achieve the parametric rate n^{-1/2} (or n^{-1/2}+m^{-1/2} for two-sample MMD) under mild conditions, even with unbounded kernels. This paper proves matching minimax lower bounds of the same order, holding on arbitrary topological spaces rather than only Euclidean space with restrictive kernels. The same lower bounds transfer immediately to estimation of the mean embedding and the centered cross-covariance operator. The result closes the optimality question for these widely used discrepancy measures.

What carries the argument

Le Cam’s two-point method applied to a carefully chosen adversarial pair of measures obtained by continuous bounded perturbations of a base measure; the construction forces the functional values to separate at rate n^{-1/2} while keeping the KL divergence between the product measures bounded.

What would settle it

Exhibit a topological space, a characteristic kernel, and a sequence of estimators whose risk is o(n^{-1/2}) uniformly over the stated class of measures, or prove that every continuous bounded function is almost surely constant for every measure in that class.

Watch

Extended reading notes

Core claim

Under the assumptions that the kernel is characteristic (or I-characteristic) and that there exists at least one probability measure admitting a non-almost-surely-constant continuous bounded perturbation, the minimax risk of estimating MMD is of exact order n^{-1/2}+m^{-1/2} and the risks of estimating HSIC and KSD are of exact order n^{-1/2}, on general topological spaces. The identical rates hold for the mean embedding and the centered cross-covariance operator.

Load-bearing premise

There must exist at least one probability measure in the class that is not almost-surely constant under every continuous bounded real function; without such a non-constant perturbation the two-point argument cannot be built.

Editorial extensions

If this is right

  • Existing U-statistic, V-statistic and accelerated estimators of MMD, HSIC and KSD are minimax optimal on far more general domains than previously known.
  • The same optimality statement holds for mean-embedding estimation and for estimation of the centered cross-covariance operator.
  • No estimator of these discrepancies can improve on the parametric rate under the stated mild conditions, even when kernels are unbounded.
  • The lower-bound technique applies uniformly across two-sample, independence and goodness-of-fit settings.

Reading between the lines

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

  • The same perturbation-plus-Le-Cam template should extend without change to other integral-probability-metric discrepancies once a characteristic property and a non-constant continuous function are available.
  • On spaces where every continuous function is constant almost everywhere (highly pathological topologies), the lower bound may fail and faster rates could become possible.
  • Practical kernel choice on non-Euclidean data (graphs, manifolds, sequences) can now safely target the parametric rate without fear that a cleverer estimator exists.
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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 / 6 minor

Summary. The paper establishes minimax lower bounds for estimating three kernel discrepancies — MMD (rate n^{-1/2} + m^{-1/2}), HSIC (n^{-1/2}), and KSD (n^{-1/2}) — on general topological spaces, plus corollaries for the mean embedding and the centered cross-covariance operator. The proofs use Le Cam's two-point method with adversarial measures built by multiplicative perturbations P^{(n)}(A) = ∫_A (1 + ε_n φ) dP_0, where ε_n = c n^{-1/2} and φ is a bounded continuous mean-zero perturbation whose existence is guaranteed by a non-degeneracy assumption (Assumptions 1(ii)/6(ii)/10(ii)). Separation of the functional values follows from injectivity of the mean embedding (characteristic / I-characteristic kernels), and the KL bound KL(P^{(n)} || P_0) ≤ α ε_n^2 is proved in Lemma B.6. The KSD result (Theorem 11) is recalled from the authors' prior AISTATS paper; the MMD, HSIC, and the two corollaries are new. Prior lower bounds were restricted to R^d with translation-invariant or radial kernels, so the generality here is the main contribution.

Significance. If the results hold — and I believe they do, modulo the framing issue in Major Comment 1 — the paper provides the first minimax lower bounds for MMD, HSIC, KSD, mean-embedding, and cross-covariance estimation that apply on general topological spaces with unbounded kernels, substantially relaxing the R^d / translation-invariant / radial assumptions of Tolstikhin et al. (2016, 2017), Chamakh and Szabó (2024), and Kalinke and Szabó (2024). The lower bounds are parameter-free in the relevant sense (the constant c in eps_n = c n^{-1/2} is arbitrary, and B > 0 is exhibited, not fitted) and rely only on characteristicness plus a mild non-degeneracy assumption. The proofs are short, checkable, and the assumptions are close to necessary for a nontrivial lower bound. This is a useful reference result for the kernel-discrepancy literature.

major comments (2)
  1. [§1, contribution (i); §3, Theorems 3, 7, 11; class P1(K;X) defined in §2] The claim in contribution (i) that the lower bounds 'match the upper ones of available estimators, hence settle their minimax optimality' for unbounded kernels is not fully supported as stated. The lower bounds are proved over P1(K;X) = {P : E_P sqrt(K(X,X)) < infty}. For unbounded K, the cited n^{-1/2} upper bounds (Kalinke et al. 2025b, 2026) require subexponentiality of sqrt(K(X,X)) under P, and even n^{-1/2} convergence of the empirical mean embedding requires the second moment E_P[K(X,X)] < infty. P1(K;X) contains measures with E[K(X,X)] = infty, over which no uniform n^{-1/2} upper bound exists. Two consequences: (a) over the stated class P1(K;X) with unbounded K, tightness at n^{-1/2} cannot be certified — the true minimax rate there may be slower; (b) a lower bound over the larger class P1 does not automatically transfer to the smaller subexponential classes where the upper bound
  2. [Abstract; §1, paragraph on convergence rates; §3.1–3.3] Related to the previous point: the abstract and §3 should state explicitly which class the optimality claim refers to. As written, Theorem 3 (and similarly 7, 11) lower-bounds the minimax risk over [P1(K;X)]^2, while the matching upper bounds cited in §1 hold only over moment/tail-restricted subclasses. A reader can currently read the paper as claiming optimality over P1(K;X) itself for unbounded kernels, which the arguments do not establish. Please align the statement of the class in the theorems, the upper-bound citations in §1, and the 'settle the question' sentence in the abstract; bounded-kernel cases (where P1 = M_1^+ and the upper bounds of Smola et al. 2007 apply) are fine as is.
minor comments (6)
  1. [§A.5] In the proof of Corollary 8 the map F is defined as F : P1(K;X) -> R, P -> C_K(P), but C_K(P) is an element of H_K; the codomain should be H_K.
  2. [§A.3] The equation label (A.9) is used both in the proof of Theorem 3 and again in the proof of Corollary 4; the second occurrence should be renumbered.
  3. [§3.1, Lemma 2] Lemma 2 (sufficient conditions for Assumption 1(ii)) assumes K in C_b(X^2), so it does not cover the unbounded-kernel regime that is the paper's main selling point. Assumption 1(ii) can in fact be verified much more cheaply whenever C_b(X) contains a non-constant function (e.g., X Tychonoff with at least two points): take phi_0 non-constant in C_b(X) and P0 a two-point mixture of Diracs at x, y with phi_0(x) != phi_0(y); finitely supported measures are always in P1(K;X). A remark to this effect would strengthen the 'mild assumptions' claim for unbounded kernels.
  4. [§1, paragraph on estimator convergence rates] In §1 the upper bounds for unbounded kernels are attributed to Kalinke et al. (2025b, 2026), which are KSD papers; please clarify which results provide the n^{-1/2} + m^{-1/2} (resp. n^{-1/2}) upper bounds for MMD (resp. HSIC) with unbounded kernels, or restrict that sentence to KSD.
  5. [§A.4] In (A.12), step (g) (C_phi > 0) uses that P^{(n)} != P0 via Lemma C.2; a pointer to Lemma C.2 at that step would parallel the treatment in (A.9) and help the reader.
  6. [Appendix A, Figure 1] The dependency chart is helpful, but 'C.5 C.3' in the header row is easy to misread; consider formatting the figure caption or layout so that external results (Appendix C) are visually distinguished from auxiliary ones (Appendix B).

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: standard Le Cam two-point construction; minor self-citation of prior KSD result is fully reproved here.

  1. self citation load bearing [p. 3 (contributions) and Theorem 11 / A.6]
    "This article expands the work of Cribeiro-Ramallo et al. (2026), which settled the minimax lower bound of KSD estimation on general topological spaces (recalled in Theorem 11, with proof for completeness)"

    The KSD lower bound is attributed to the authors' own prior paper. This is only a minor self-citation: the complete Le Cam argument is written out in A.6 under Assumption 10, so the present manuscript does not rely on an external unverified claim. Score contribution is therefore minimal.

full rationale

The claimed minimax rates are obtained by the classical Le Cam method: adversarial pairs are built by continuous bounded perturbations of a base measure (A.8, A.16) with free scale ε_n = c n^{-1/2} (c > 0 arbitrary), separation of the functional is shown to be Θ(ε_n) via injectivity/characteristicness plus the mean-embedding identity (Lemma B.5), and the product KL is O(1) by the second-order expansion of KL under small perturbations (Lemma B.6). Balancing separation against the KL budget forces the n^{-1/2} (resp. n^{-1/2}+m^{-1/2}) lower bound; nothing is fitted to data, and no equation equates the target rate to an input by definition. The only self-reference is that Theorem 11 restates the authors' earlier KSD lower bound; the full proof is reproduced in A.6 under the same mild assumptions, so the citation is not load-bearing. Corollaries for the mean embedding and centered cross-covariance follow by the reverse triangle inequality from the same pairs and are likewise self-contained. No self-definitional loop, fitted-as-prediction step, uniqueness import, or renamed empirical pattern appears.

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

The central claims rest on standard measure-theoretic and RKHS facts plus three domain assumptions (characteristic / I-characteristic kernels and existence of a non-constant continuous perturbation). No free parameters are fitted to data; the constants c, c1, c2 that scale the perturbations are arbitrary positive numbers chosen for the Le Cam argument. No new physical or statistical entities are postulated.

assumptions (5)
  • standard math Le Cam's two-point method (Theorem C.5 / Cam 1973; Tsybakov 2009): if two parameters are 2s-separated and their laws have KL ≤ α, then any estimator has risk at least f(α)s with positive probability.
    Invoked as the sole lower-bound engine in the proofs of Theorems 3, 7 and 11.
  • domain assumption Kernel K is characteristic (injective mean embedding) on P1(K;X) for MMD; product kernel is I-characteristic for HSIC; Stein kernel is characteristic w.r.t. P0 for KSD.
    Assumptions 1(i), 6(i), 10(i). Converts positive embedding distance of the adversarial pair into positive discrepancy; without it the distance control step fails.
  • domain assumption Existence of (P0, φ0) ∈ P1 × Cb(X) with φ0 not P0-a.s. constant (and the analogous product-space version).
    Assumptions 1(ii), 6(ii), 10(ii). Supplies the continuous perturbation used to build the adversarial measures via Lemma C.1.
  • standard math Bochner integrability of the canonical feature map under P ∈ P1(K;X); separability of the Stein RKHS when needed for measurability.
    Background RKHS/measure theory used to define mean embeddings and KSD (Section 2).
  • standard math KL of product measures factors as the sum of KLs (Lemma C.3 / Tsybakov).
    Used to bound KL(P_n^{(n)} ∥ P_0^n) by n times the single-copy KL.

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Pith. "Pith review of Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD." pith.science (2026). https://pith.science/paper/2WBBHXPP

@misc{pith2026260724235,
  author       = {Pith},
  title        = {Pith review of: Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WBBHXPP}},
  note         = {Machine review of arXiv:2607.24235}
}
abstract

Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.

Figures

Figures reproduced from arXiv: 2607.24235 by the authors.

Figure 1
Figure 1. Summary of the dependencies of our results. R1 ← R2 means that “result R1 depends on R2”. B.3, B.6, C.5 and C.3 are all used in Theorems 3, 7 and 11. A.1 Proof of Lemma 2 Let φ0 ∈ HK ⊆ Cb(X ) be the assumed non-constant element, where the inclusion HK ⊆ Cb(X ) holds by K ∈ Cb(X 2 ) (Steinwart and Christmann, 2008, Lemma 4.28). Further, as X is separable, there exists a countable dense subset {xi}∞ i=1 ⊆ X . Let P0 =… view at source ↗

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Pith tools

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