The discrepancy in min-max statistics between two random matrices with finite third moments

Chengfu Wei; Yiming Chen; Zijun Chen

REVIEW 3 major objections 4 minor 18 references

The discrepancy in min-max statistics between two random matrices with finite third moments

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves a quantitative Gordon-type comparison inequality for min-max statistics of two random matrices, requiring only finite absolute third moments.

desk verdict A promising non-Gaussian min-max comparison idea, but the final Strassen step proves a coupling, not the independent-pair tail the theorem claims. read the letter →

arxiv 2411.08303 v1 pith:UYOGAXSZ submitted 2024-11-13 math.PR

Zijun Chen , Yiming Chen , Chengfu Wei This is my paper · ORCID

classification math.PR MSC 60F0560G1560G7060H05

keywords min-maxstatisticsrandommatricesStein'smethodexchangeablepairsGaussianapproximationvectorscouplingquantitativeGordoninequality

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 proves a quantitative comparison inequality for the min-max statistics of two independent random matrices. The only conditions on the entries are zero mean and finite absolute third moments, replacing the Gaussian assumption required by earlier Gordon-type bounds. The theorem gives an explicit upper bound on the probability that the row-minimum/column-maximum statistics of the two matrices differ by more than a threshold set by the parameters $\beta$, $\delta$, and $\tau$. The bound is expressed through the fluctuation of entry products, the maximum third moment of entries, and the difference of the covariance structures. This matters because min-max statistics appear throughout random matrix theory, optimization, and empirical process theory, where Gaussian assumptions are often too restrictive.

What carries the argument

The key machinery is the smooth 'soft min-max' function $F_{\beta\delta}(x) = -\frac{1}{\beta\delta} \log \sum_{i=1}^n \left(\sum_{j=1}^m \exp(\beta x_{ij})\right)^{-\delta}$, which approximates $\min_i \max_j x_{ij}$ within a window of width $\frac{1}{\beta}\log m + \frac{1}{\beta\delta}\log n$. The proof's core is Lemma 2.5, which shows that for any thrice-smooth $g$, the sums of the second and third derivatives of $g \circ F_{\beta\delta}$ over all index pairs and triples are bounded by $\beta$, $\delta$, and the derivative norms of $g$. These bounds let the expectation of $g \circ F_{\beta\delta}(X)$ be compared to that of $g \circ F_{\beta\delta}(X')$ through two Stein-type steps: an exchangeable-pair identity that handles the non-Gaussian difference from a matched Gaussian $Y$, and a Gaussian interpolation formula that bounds the difference between the two Gaussian bridges $Y$ and $Y'$.

What would settle it

For a concrete non-Gaussian ensemble such as i.i.d. Rademacher entries, compute by simulation the probability that the min-max statistics of two independent copies differ by more than the threshold, and compare it to the bound for several $\beta$, $\delta$, $\tau$; a single violation for valid parameters would refute the theorem.

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Extended reading notes

Core claim

The central discovery is that two independent $n \times m$ random matrices $X$ and $X'$ with mean-zero entries and finite absolute third moments have min-max statistics that are close in distribution in a quantitatively controlled way. Specifically, for every $\beta>0$, $\delta>0$, and $\tau>1/(\beta(1+\delta))$, the probability that $|\min_i \max_j X_{ij} - \min_i \max_j X'_{ij}|$ exceeds $2(\log n/(\beta\delta) \vee \log m/\beta)+3\tau$ is at most $\varepsilon + C\beta(1+\delta)\tau^{-1}(B_1+B_1'+B_3+\beta(1+\delta)(B_2+B_2'))/(1-\varepsilon)$, where $\varepsilon=\sqrt{e^{-\alpha}(1+\alpha)}<1$ with $\alpha=\beta^2(1+\delta)^2\tau^2-1$, and $B_1,B_2,B_3$ are explicit data-dependent quantities. In the Gaussian case the bound simplifies to $\varepsilon+C\beta(1+\delta)\tau^{-1}B_3/(1-\varepsilon)$. The result is obtained by smoothing the non-smooth min-max map with a log-sum-exp free-energy function and then comparing expectations through Stein's method with exchangeable pairs and Gaussian interpolation.

Load-bearing premise

The argument assumes that the Stein helper function $h$ built from the smoothed min-max $f$ inherits the same derivative-sum bounds as $f$, a step that is asserted rather than proved in the proof of Theorem 2.8.

Editorial extensions

If this is right

The min-max statistics of any two mean-zero random matrices with finite third moments are close in distribution, with a probability tail that decays at least like $\tau^{-1}$.
Choosing $\beta$, $\delta$, $\tau$ appropriately recovers the Gaussian quantitative Gordon inequality as the Gaussian special case.
The result holds even when entries within a matrix are dependent, since only moment conditions on the entries are assumed.
The bound depends only on $B_1$, $B_2$, $B_3$, which are in principle computable or estimable from the entry distributions.

Reading between the lines

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

The paper does not explore, but the same smoothing-plus-Stein strategy could in principle apply to other non-smooth functionals of random matrices, such as the spectral norm or the largest eigenvalue, whenever a soft approximation with controlled derivative sums exists.
The choice of $\beta$ and $\delta$ is left open; an optimal tuning might yield sharper effective bounds for specific entry distributions, but the paper does not investigate this.
The argument suggests that the third-moment quantities $B_1$, $B_2$, $B_3$ are the natural measures of discrepancy; one could test empirically whether they are also necessary by constructing distributions with finite second but infinite third moments where the bound should fail.

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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

The paper has a genuine new idea: a quantitative min-max comparison for matrices with only finite third moments, obtained by combining exchangeable-pair Stein methods with a smooth free-energy approximation. The derivative computations for F are careful, and the borrowing from Chernozhukov-Chetverikov-Kato and Meckes is sensible. If the goal were a coupling inequality, the strategy would be close to right.

But the theorem as stated is not proved, and in fact it cannot be true. The proof ends by showing, for every Borel set A, that P(minmax X in A) is bounded by P(minmax X' in A^{2lambda+3tau}) plus an error term. Strassen's lemma (Lemma 2.1) turns that distributional inequality into the existence of some coupling (V,W) with P(|V-W|>delta) small. It says nothing about the original independent pair X,X'. For Bernoulli(1/2) marginals, the distributional condition holds with delta=0.5 and epsilon=0, while two independent copies differ with probability 1/2. So the final 'By Lemma 2.1' step is invalid. The theorem must be restated as a coupling or distributional comparison, or the tail probability must refer to the coupled pair, not the independent matrices.

Separately, after (2.3) a factor (1-epsilon)^{-1} is dropped: the chain should give (1-epsilon)^{-1} E[g(F(X'))], and then the epsilon from Lemma 2.4 should appear as epsilon/(1-epsilon). That is a minor algebraic slip, but it needs fixing. The application of Lemma 2.5 to the Poisson-solver function h is also not justified: h is an integral of f, and one has to check that the derivative-sum bounds survive the integration. This is likely repairable by differentiating under the integral and integrating the t-powers, but it is not in the paper. Finally, the Gaussian corollary in Theorem 1.2 drops the B1 and B2 terms, which do not vanish for Gaussian matrices; that statement as written is also false.

Specialists in high-dimensional probability who care about quantitative Gordon-type inequalities would find the intended result useful, but the main theorem is unsupported as written. I would not accept it. If the authors restate the theorem, fix the algebra, and prove the h bounds, the paper could become publishable. It deserves a careful referee's time in the sense that a referee could guide that repair, but it should not be published in this form.

Referee Report

3 major / 4 minor

Summary. The paper claims a quantitative comparison of min-max statistics of two independent n x m random matrices with mean zero and finite absolute third moments. Theorem 1.2 asserts a tail bound on |min_i max_j X_ij - min_i max_j X'_ij| in terms of the moment quantities B1, B1', B2, B2', B3 and a constant epsilon defined in (1.1). The proof smooths the non-smooth min-max functional through the function F_{beta,delta}, approximates indicators by a smooth function via Lemma 2.4, compares expectations E[f(X)] and E[f(X')] using exchangeable pairs and Gaussian interpolation in Theorem 2.8, and concludes by invoking Lemma 2.1 (Strassen's theorem).

Significance. If valid, the result would be a substantial non-Gaussian extension of the quantitative Gordon/Sudakov-Fernique inequalities obtained by Peccati and Turchi, with explicit dependence on third-moment quantities. The smoothing construction and the decomposition into B1, B2, B3 are natural and the constants are explicit rather than fitted. However, the proof as written has a load-bearing gap in the final Strassen step and an unjustified application of Lemma 2.5 to the Stein helper function h, so the main theorem is not established.

major comments (3)

[Section 2, proof of Theorem 1.2, final sentence] The final inference 'By Lemma 2.1, we complete our proof' is invalid. Lemma 2.1 (Strassen) states that the distributional inequality mu(A) <= nu(A_delta) + epsilon is equivalent to the existence of some coupling (V,W) on a common probability space with P(|V-W| > delta) <= epsilon. The chain preceding it establishes, at most, such a distributional inequality for the laws of minmax(X) and minmax(X'). It does not control P(|minmax(X) - minmax(X')| > ...) for the independent pair X,X' stated in Theorem 1.2. As a concrete obstruction, if mu = nu = Bernoulli(1/2) and delta = 1/2, the distributional condition holds with epsilon = 0, while two independent copies differ by 1 with probability 1/2. Thus the theorem as stated is not a consequence of the preceding argument; at best the proof supports a different coupling statement.
[Equations (2.3) and the chain after (2.10)] There is an algebraic slip in the chain following Theorem 2.8. From (2.3), P(minmax(X) in A) <= (1-epsilon)^{-1} E[g(F(X))]. Combining this with (2.10) gives (1-epsilon)^{-1} E[g(F(X'))] + C beta (1+delta) tau^{-1}(...)/(1-epsilon), not E[g(F(X'))] + C beta (1+delta) tau^{-1}(...)/(1-epsilon) as written. Applying the upper bound for g from Lemma 2.4 then yields an epsilon/(1-epsilon) term rather than epsilon, and the claimed constants in Theorem 1.2 are not supported as stated.
[Proof of Theorem 2.8, paragraphs after Lemma 2.6] Lemma 2.5 gives derivative-sum bounds for h0 = g composed with F_{beta,delta}, namely (2.4) and (2.5) in terms of ||g'||, ||g''||, and ||g'''||. In the proof of Theorem 2.8, these bounds are instead applied to the function h defined in Lemma 2.6, the Stein/Poisson-solver helper for f = g composed with F_{beta,delta}. The proof does not show that h inherits the needed bounds on the sums of its second and third derivatives from f; in particular, the estimates 'by (2.4)' and 'by (2.5)' in the bounds for the B1 and B2 terms are unjustified. This is a substantive technical gap, not a typo, because h is defined via an integral involving f(sqrt(t)x + sqrt(1-t)Y) and its derivatives are not directly the same as those of f.

minor comments (4)

[Throughout] There are several typos: 'Chaterjee' should be 'Chatterjee', 'whcih' should be 'which', 'borel' should be 'Borel', and 'Elizabeth' should be 'Meckes'.
[Proof of Theorem 1.2, displayed chain] The line 'P(g composed with F(X') in A_{lambda+3tau}) + epsilon + ...' is not the correct intermediate step: Lemma 2.4 gives E[g(F)] <= epsilon + (1-epsilon) P(F in A_{lambda+3tau}), so the intermediate probability should involve F(X') in A_{lambda+3tau}, not g(F(X')) in that set. The subsequent final bound on minmax(X') is consistent with the corrected version, but the displayed chain should be repaired.
[Equation (2.9)] The bound on the second-derivative sum in the Gaussian interpolation step is attributed to '(2.5)', but the relevant second-derivative bound is (2.4). This is a citation typo that should be corrected.
[Lemma 2.2 proof] The proof of Lemma 2.2 introduces beta' and then identifies beta' = beta delta at the end; the intermediate notation is confusing and the displayed inequalities would benefit from a clearer separation of the two approximation steps.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bound is expressed in explicit functionals of fixed moments and uses external lemmas; no input is equivalent to the target by construction.

full rationale

The paper's central estimate is a quantitative inequality whose right-hand side consists of explicit functionals of the fixed distributions of X and X' (B1, B1', B2, B2', B3) and the smoothing parameters beta, delta, tau. None of these quantities is fitted to the target min-max discrepancy; B1, B2, B3 are unconditional moment-type expectations, and epsilon in (1.1) is an explicit function of beta, delta, tau only. The proof chain invokes external tools -- Strassen's theorem, Kato's smooth approximation, Meckes' Poisson representation, and Gaussian interpolation -- and does not assume the conclusion or a renamed version of it. There are no load-bearing self-citations: the cited results are from external authors (Kato, Chatterjee-Meckes, Van Handel, Vershynin, etc.), not from the present authors' prior work, and none is used to forbid alternatives. The reader's and skeptic's concerns about the final use of Strassen's lemma and about whether the helper function h inherits the derivative-sum bounds are potential correctness or proof-gap issues, not circularity: they concern whether an intermediate inequality is established, not whether the theorem reduces to its own assumptions by definition or by fitted inputs. Under the stated circularity criteria, no circular step is present.

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

The central claim rests on standard external lemmas: Strassen's coupling theorem, the Chernozhukov-Chetverikov-Kato smoothing lemma, Meckes' Poisson-solver representation, and the Gaussian interpolation formula. In addition, the proof uses an unstated technical assumption that the Poisson-solver h inherits the derivative-sum bounds proved for f; this is not established. No parameters are fitted to data and no new entities are introduced.

assumptions (5)

standard math Strassen's theorem as stated in Lemma 2.1 characterizes existence of couplings via an inequality of measures.
Used to convert the distributional comparison into a coupling of min-max statistics; the theorem is cited and not proved.
standard math The smoothing lemma of Chernozhukov-Chetverikov-Kato (Lemma 2.4 in the paper) provides a thrice differentiable g with the stated two-sided bound and derivative bounds.
Assumed as an external result; it is the source of epsilon and of the tau^{-1} factors.
standard math Meckes' representation (Lemma 2.6) expresses f(x)-E[f(Y)] through a Poisson-solver h.
External lemma used for exchangeable-pair approximation.
standard math Gaussian interpolation formula (Lemma 2.7) gives E[p(xi)-p(eta)] via the covariance difference.
External lemma used to bound E[f(Y)-f(Y')].
ad hoc to paper The Poisson-solver h inherits from f the derivative-sum bounds of Lemma 2.5.
Not proved in the paper but required in the proof of Theorem 2.8 when Lemma 2.5 is applied to derivatives of h.

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Cite this review

Pith. "Pith review of The discrepancy in min-max statistics between two random matrices with finite third moments." pith.science (2026). https://pith.science/paper/UYOGAXSZ

@misc{pith2026241108303,
  author       = {Pith},
  title        = {Pith review of: The discrepancy in min-max statistics between two random matrices with finite third moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYOGAXSZ}},
  note         = {Machine review of arXiv:2411.08303}
}

read the original abstract

We propose a novel coupling inequality of the min-max type for two random matrices with finite absolute third moments, which generalizes the quantitative versions of the well-known inequalities by Gordon. Previous results have calculated the quantitative bounds for pairs of Gaussian random matrices. Through integrating the methods utilized by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation, this study eliminates the Gaussian restriction on random matrices, enabling us to achieve more extensive results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

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Pith tools

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

[1] [1]

An error bound in the Sudakov-Fernique in equality

Chatterjee, S. An error bound in the Sudakov-Fernique in equality. arXiv preprint math/0510424 (2005)

arXiv 2005

[2] [2]

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Chatterjee, S. and Elizabeth M. Multivariate normal app roximation using exchangeable pairs. arXiv preprint math/0701464 (2007)

work page Pith review arXiv 2007

[3] [3]

Chetverikov, D

Chernozhukov, V. Chetverikov, D. and Kato, K. Gaussian a pproximation of suprema of empirical processes. (2014): 1564-1597

work page 2014

[4] [4]

M., Conze, J.P

Fernique, X. M., Conze, J.P. Gani, J. Fernique, X. Regula rit´ e des trajectoires des fonctions al´ eatoires gaussiennes. Springer Berlin Heidelberg, (1975)

work page 1975

[5] [5]

Some inequalities for Gaussian processes and applications

Gordon, Y. Some inequalities for Gaussian processes and applications. Israel Journal of Mathematics 50 (1985): 265-289

work page 1985

[6] [6]

Elliptically contoured distributions

Gordon, Y. Elliptically contoured distributions. Prob ability theory and related ﬁelds 76.4 (1987): 429-438

work page 1987

[7] [7]

Une in´ egalit´ e du type de Slepian et Gordon sur les processus gaussiens

Kahane, J.P. Une in´ egalit´ e du type de Slepian et Gordon sur les processus gaussiens. Israel Journal of Mathematics 55 (1986): 109-110

work page 1986

[8] [8]

On Stein’s method for multivariate normal app roximation

Meckes, E. On Stein’s method for multivariate normal app roximation. High dimensional probability V: the Luminy volume. Vol. 5. Institute of Mathe matical Statistics, (2009). 153- 179

work page 2009