Operator $\ell_p\to\ell_q$ norms of Gaussian matrices

Marta Strzelecka; Rafa{\l} Lata{\l}a

Operator $\ell_p\to\ell_q$ norms of Gaussian matrices

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Pith's one-line read For any variance profile, the expected $\ell_p\to\ell_q$ norm of a Gaussian matrix equals, up to $p,q$-only constants, the sum of the largest row $\ell_{p^*}$-norm, the largest column $\ell_q$-norm, and the expected largest Gaussian entry.

desk verdict Resolves the Guédon–Hinrichs–Litvak–Prochno conjecture for Gaussian matrices with arbitrary variance profile, and the proof holds together. read the letter →

arxiv 2502.02186 v3 pith:4T6R3OPJ submitted 2025-02-04 math.PR math.FA

Rafa{l} Lata{l}a , Marta Strzelecka This is my paper · ORCID

classification math.PRmath.FA MSC 60B2046B09

keywords Gaussianrandommatricesoperatornormsell_ptoell_qvarianceprofiletwo-sidedboundshypercontractivitymatrixtheoryconjectureconfirmation

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 establishes a complete two-sided estimate for the expected operator norm of a Gaussian random matrix whose entry variances are arbitrary (a 'variance profile'): acting from $\ell_p$ to $\ell_q$ with $1\le p\le 2\le q\le \infty$, the norm is determined, up to constants depending only on $p$ and $q$, by the three quantities: the maximum $\ell_{p^*}$ norm of a row, the maximum $\ell_q$ norm of a column, and the expected maximum of the Gaussian entries themselves. This confirms a conjecture of Guédon, Hinrichs, Litvak, and Prochno from 2017, previously known only in the cases $p=1$, $q=\infty$, and $p=q=2$. The proof supplies a new, non-spectral argument for the already-known $p=q=2$ spectral case. The result matters because matching two-sided bounds of this type automatically yield moment and tail estimates, and characterize when an infinite Gaussian matrix is almost surely a bounded operator between $\ell_p$ and $\ell_q$.

What carries the argument

The argument runs on three main mechanisms. First, the target expression is shown to be equivalent to several other forms: the expected maximum of Gaussian entries is comparable to an inf-sup quantity $\max_{k}\inf_{|I|=k}\sqrt{\log k}\,\max_{(i,j)\notin I}|a_{ij}|$, and to the expected $\ell_{p^*}$ and $\ell_q$ norms of the rows and columns of $G_A$ (Theorem 2). Second, the upper bound is proved by splitting the norm into a part controlled by a standard net argument with a $\sqrt{\log(mn)}$ error (Proposition 15) and a part controlled by a block-diagonal decomposition with a weighted-supremum error (Proposition 16); a combinatorial lemma (Proposition 18) then shows that these two bounds force the logarithmic factor to disappear. Third, in the most delicate regime $p^*,q\in[2,3]$, the proof introduces an interpolation norm $\|x\|_{2,q,\lambda}$ on $\mathbb{R}^m$ whose key property—hypercontractivity of Gaussian variables (Lemma 24)—allows a multiscale induction; the Slepian–Fernique comparison (Lemma 36) bounds the sup over small-coordinate vectors. The Gaussian assumption enters precisely at these three points: hypercontractivity, Gaussian concentration, and comparison.

What would settle it

For fixed $p=4/3$, $q=4$ and the variance profile $a_{ij}=(i+j)^{-1}$ on $m\times n$ grids of increasing size, estimate $\mathbb{E}\|G_A\|_{p\to q}$ by Monte Carlo; the theorem predicts the ratio to $\max_i\|(a_{ij})_j\|_{p^*}+\max_j\|(a_{ij})_i\|_q+\mathbb{E}\max_{i,j}|a_{ij}g_{ij}|$ stays within a $p,q$-dependent constant, so a systematically growing ratio would refute the claim.

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Extended reading notes

Core claim

The central claim is Theorem 2: for every deterministic $m\times n$ matrix $A=(a_{ij})$ and every $1\le p\le 2\le q\le \infty$, $\mathbb{E}\|(a_{ij}g_{ij})_{i,j}:\ell_p^n\to\ell_q^m\|$ is comparable, with constants depending only on $p$ and $q$, to $\max_i\|(a_{ij})_j\|_{p^*}+\max_j\|(a_{ij})_i\|_q+\mathbb{E}\max_{i,j}|a_{ij}g_{ij}|$, where $g_{ij}$ are iid standard Gaussians and $p^*$ is the Hölder conjugate of $p$. Equivalent forms replace the expected maximum by an inf-sup over deleting $k$ rows and columns with a $\sqrt{\log k}$ factor, and by expected row/column maxima of the randomized matrix. The authors obtain the upper bound by combining a dimension-dependent bound with a logarithmic factor (Proposition 15) and a block-decomposition bound (Proposition 16), then eliminating the logarithm through a graph-theoretic exponent reduction. In the range $p^*,q\in[2,3]$ the proof uses Gaussian hypercontractivity in an interpolated norm and a Slepian–Fernique comparison; the Gaussian character of the entries is used essentially throughout.

Load-bearing premise

The proof relies essentially on the entries being Gaussian: hypercontractivity, Gaussian concentration, and the Slepian–Fernique comparison each enter at a different stage, and the authors note in Remark 9 that replacing Gaussians by symmetric Bernoulli entries changes the behavior of the third term.

Editorial extensions

If this is right

Corollary 10: the two-sided bound for the expectation lifts to all moments and to Gaussian tail bounds, so for every $\rho\ge1$, $(\mathbb{E}\|G_A\|_{p\to q}^{\rho})^{1/\rho}$ is comparable to the same three-term expression plus $\sqrt{\rho}\,\max_{i,j}|a_{ij}|$.
Corollary 5: an infinite Gaussian matrix $(a_{ij}g_{ij})$ defines a bounded operator on $\ell_p\to\ell_q$ almost surely if and only if the three quantities are finite; this is a sharp, checkable condition.
Corollaries 7 and 8: the same three-term formula holds for Gaussian mixtures and for independent symmetric Weibull entries with shape parameter $r\le2$ (with $\sqrt{\log k}$ replaced by $(\log k)^{1/r}$); for the all-ones matrix this recovers the known Weibull estimates.
Corollary 13: a general upper bound for independent centered entries with no Gaussian assumption follows, giving a dimension-dependent bound in terms of $\ell_{p^*}$ and $\ell_q$ moments.
The $p=q=2$ spectral case now has a proof that avoids spectral theory and the trace method, replacing it with interpolation and comparison techniques.

Reading between the lines

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

The paper leaves open the sharp dependence on $p$ and $q$; for the all-ones matrix the correct growth is $\sqrt{p^*\wedge\log n}+\sqrt{q\wedge\log m}$, but the constants in Theorem 2 grow like $(p^*\vee q)^{13/2}$ or $(p^*\vee q)^{5/2}$. One can test the refined conjecture numerically for small $p^*,q$ by computing the ratio for the all-ones matrix.
The Bernoulli case ($r=\infty$) is explicitly excluded, and the paper conjectures a different three-term formula with a hard-to-compute third term; if that conjecture is true, the Gaussian-versus-Bernoulli gap is exactly the distinction between $\sqrt{\log k}$ and $\log k$ scaling for the large-entries term.
The graph-theoretic exponent-reduction technique (Propositions 27–29) is not specific to Gaussian entries; it may extend to other log-concave or sub-exponential profiles, with the hypercontractivity step replaced by the appropriate moment bound.
Corollary 5 suggests a probabilistic proof that almost-sure boundedness of Gaussian operators between $\ell_p$ spaces has a deterministic characterization; this could be relevant for the theory of random linear operators on sequence spaces.

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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 settles a 2017 conjecture: for deterministic A and 1 ≤ p ≤ 2 ≤ q ≤ ∞, the expected ℓ_p→ℓ_q norm of (a_ij g_ij) is two-sided comparable, up to p,q-dependent constants, to the sum of the three natural terms. That is the headline, and it is a genuine result, not a parameter scan. The novelty is real: the full range p* , q ≥ 2 was open, with only the endpoints p=1 or q=∞ and the spectral case p=q=2 known. The proof of the p=q=2 case without spectral theory is a nice extra, and the dimension-free two-sided bounds imply moment and tail estimates that go beyond the expectation statement.

The paper is well organized and the authors are honest about the limits of their constants: they do not claim the optimal dependence on p and q, and they explicitly flag where the constant blows up. The main theorem is deduced cleanly from Propositions 15 and 16 via Proposition 18, and the proofs of those propositions are detailed. I did not machine-check every algebraic step, but the logic is coherent and the dependencies are transparent.

Soft spots, in proportion: the most deferred step is Lemma 36, the Slepian–Fernique comparison quoted from van Handel's paper. Proposition 35, and hence the p* , q ∈ [2,3] case including p=q=2, rests on it. That is a published standard tool, so it is not an actual gap, but it is the part a referee should look at most carefully. Also, the Gaussian assumption is load-bearing, not a weakness: the authors rightly show the Weibull (r≤2) extension and note in Remark 9 that Bernoulli entries behave differently. The constants in the upper bound are non-optimal in p,q; the authors say so and even conjecture the sharp form. That is honest, not a flaw.

Who is this for? Anyone working on random matrices with variance profiles, operator norms of structured Gaussian matrices, or dimension-free estimates in Banach space theory. It deserves a serious referee: the proof is intricate but the structure is sound, and the conjecture resolution is important enough to justify careful checking. I would accept it for peer review and recommend moving forward after the standard verification of the quoted Slepian–Fernique step.

Referee Report

0 major / 3 minor

Summary. The paper studies the expected ℓ_p^n → ℓ_q^m operator norms of Gaussian matrices with a deterministic variance profile (a_ij^2). The main result, Theorem 2, proves that for every 1 ≤ p ≤ 2 ≤ q ≤ ∞ and every deterministic matrix A, E||G_A||_{p→q} is comparable, up to constants depending only on p and q, to max_i ||(a_ij)_j||_{p*} + max_j ||(a_ij)_i||_q + E max_{i,j} |a_ij g_ij|. This confirms Conjecture 1 of Guédon–Hinrichs–Litvak–Prochno (2017), previously known only for p=1 or q=∞ and for the spectral case p=q=2. The proof is split into two upper estimates (Propositions 15 and 16) that are combined through a deterministic block decomposition (Proposition 18). Along the way the authors reprove the p=q=2 case without spectral-theoretic tools. The paper also derives a boundedness criterion for infinite Gaussian matrices, moment and tail bounds, and an extension to symmetric Weibull entries with shape parameter r ≤ 2, with a remark explaining why r > 2, in particular the Bernoulli case, behaves differently.

Significance. The result, if correct, resolves a well-known conjecture in the theory of structured random matrices and gives the first sharp two-sided bounds for these operator norms outside the extremal cases. One notable strength is the proof architecture: the high-level reduction (Theorem 2 from Propositions 15, 16, and 18) is clean and each proposition is proved in detail; constants are reported honestly, including the explicit dependence on p and q in Remark 3. The paper is also careful about scope: it proves the Gaussian and Weibull (r ≤ 2) cases and explicitly flags in Remark 9 that the Bernoulli limit diverges from the stated form, which is a falsifiable and testable prediction. The main proof uses standard probabilistic tools (Gaussian concentration, hypercontractivity, Slepian–Fernique comparison); the only deferred ingredient is the comparison lemma (Lemma 36), which is quoted from a published proof in [18], but this is a standard reference rather than a gap. I see no internal inconsistency and no circularity: the p=q=2 case is reproved from the new machinery, not assumed.

minor comments (3)

[Section 3.2, Proposition 27] The exponent in the logarithmic factor is typeset confusingly ("Logγ3 − ln((γ1p∗) ∨ (γ2q)))"); it should be written unambiguously, for example as Log^{γ3 / (-ln((γ1 p_*) ∨ (γ2 q)))} or with an explicit lower bound on k0, so that the dependence on the parameters is clear to the reader.
[Section 3.1, Proposition 20] The notation "γ q∗" in the bound "√γq∗ Logm" is ambiguous; the proof uses the sparsity bound |{i : |s_i| ≥ γ^{-1}}| ≤ γ^{q_*}, so the statement should display γ^{q_*} as an exponent to avoid confusion with a product γ·q_*.
[Throughout] The arXiv source contains numerous OCR artifacts (e.g., "/BD" for indicators, "/greaterorsimilar" for ≳, and broken spacing in the title and abstract); these should be cleaned in the final version, since they distract from the otherwise clear presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained relative to published, independently established ingredients.

full rationale

The main theorem is derived from Propositions 15 and 16 through the deterministic block-decomposition Proposition 18; no step assumes the Guédon–Hinrichs–Litvak–Prochno conjecture. The harder upper bound in the range p*,q in [2,3] rests on Proposition 35, whose proof imports the Slepian–Fernique comparison from van Handel [18, proof of Theorem 4.1]; this is an external published theorem used as a lemma, not an assumption of the conclusion being proved. The equivalences among the candidate expressions are cited to [1, Section 5.4], which is a published proof of a statement about Gaussian maxima, not a proof of the operator-norm comparability itself; it is used only to rewrite the claimed bound, and the norm inequality is established independently by Propositions 15, 16, 18, 23, 27, and 34. The paper's own authors appear in several cited works, but those citations are to published auxiliary results with proofs (moment equivalences, maximal-entry estimates, Weibull moment comparison), not to an unverified uniqueness claim or to a restatement of the theorem. No parameter is fitted to data, no quantity called a prediction is the input in disguise, and the p=q=2 case is reproved with the same tools rather than assumed. The Gaussian assumption is the intended scope, explicitly contrasted with the Bernoulli regime in Remark 9. Accordingly, the derivation chain does not reduce to its own inputs.

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

The central claim rests on standard probabilistic inequalities (Gaussian hypercontractivity, concentration, Slepian-Fernique) and on previously published endpoint results that are part of the literature. No ad hoc assumptions, fitted parameters, or new postulated entities are introduced.

assumptions (5)

standard math Gaussian hypercontractivity: for iid standard Gaussians g_i and q ≥ 2, (E|∑ c_i g_i|^q)^{1/q} ≤ √q (∑ c_i^2)^{1/2}.
Used in Lemma 24 (inequality 27) and Lemma 21 to bound expected norms of Gaussian vectors.
standard math Gaussian concentration inequality for Lipschitz functions of standard Gaussian vectors.
Used in Proposition 18, Lemma 22, and Lemma 33 via [3, Theorem 5.6].
standard math Slepian-Fernique comparison lemma for centered Gaussian processes.
Used in Lemma 36 to dominate the expected supremum of the bilinear form by a sum of simpler Gaussian processes.
domain assumption Known two-sided bounds for p=1 or q=∞ from [7, Remark 1.4] and [1, Propositions 1.8 and 1.10].
Used to reduce the proof to the case p*,q ∈ [2,∞) in the introduction.
domain assumption Equivalence between E max_{i,j} |a_ij g_ij| and the combinatorial quantity max_{k≥0} inf_{|I|=k} √Log k max_{(i,j)∉I} |a_ij|, proven in [18, Lemmas 2.3 and 2.4].
Used in Theorem 2 and Corollary 8, equation (13).

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Cite this review

Pith. "Pith review of Operator $\ell_p\to\ell_q$ norms of Gaussian matrices." pith.science (2026). https://pith.science/paper/4T6R3OPJ

@misc{pith2026250202186,
  author       = {Pith},
  title        = {Pith review of: Operator $\ell_p\to\ell_q$ norms of Gaussian matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4T6R3OPJ}},
  note         = {Machine review of arXiv:2502.02186}
}

abstract

We confirm the conjecture posed by Gu\'edon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}\|(a_{ij}g_{ij})_{i\le m, j\le n}\colon \ell_p^n \to \ell_q^m\|$ is comparable, up to constants depending only on $p$ and $q$, to \[ \max_i \|(a_{ij})_j\|_{p^*} +\max_j \|(a_{ij})_i\|_{q} +\mathbb{E} \max_{i,j} |a_{ij}g_{ij}| \] provided that $1\le p \le 2\le q \le \infty$. This was known before only in the case $p=1$ or $q=\infty$, and in the spectral case $p=2=q$. We also reprove the conjecture in the case $p=2=q$ without using spectral theory (which was employed in the previously known proof).

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Works this paper leans on

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [18]

van Handel, On the spectral norm of Gaussian random matrices , Trans

R. van Handel, On the spectral norm of Gaussian random matrices , Trans. Amer. Math. Soc. 369 (2017), no. 11, 8161–8178. MR 3695857 19. , Structured random matrices , Convexity and concentration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 107–156. MR 3837269 University of W arsa w, Institute of Mathematics, Banacha 2, 02–097 W arsa w, P...

work page 2017

[2] [1]

Adamczak, J

R. Adamczak, J. Prochno, M. Strzelecka, and M. Strzelecki , Norms of structured random matrices, Math. Ann. 388 (2024), no. 4, 3463–3527. MR 4721775

work page 2024

[3] [2]

A. S. Bandeira and R. van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries , Ann. Probab. 44 (2016), no. 4, 2479–2506. MR 3531673

work page 2016

[4] [3]

Boucheron, G

S. Boucheron, G. Lugosi, and P. Massart, Concentration inequalities, Oxford University Press, Oxford, 2013, A nonasymptotic theory of independence, With a foreword by Michel Ledoux. MR 3185193

work page 2013

[5] [4]

Brailovskaya and R

T. Brailovskaya and R. van Handel, Extremal random matrices with independent entries and matrix superconcentration inequalities, preprint, arXiv:2401.06284, (2024)

arXiv 2024

[6] [5]

V. H. de la Peña and E. Giné, Decoupling. from dependence to independence , Springer-Verlag, New York, 1999

work page 1999

[7] [6]

Eskenazis, P

A. Eskenazis, P. Nayar, and T. Tkocz, Gaussian mixtures: entropy and geometric inequalities , Ann. Probab. 46 (2018), no. 5, 2908–2945. MR 3846841

work page 2018

[8] [7]

Guédon, A

O. Guédon, A. Hinrichs, A. E. Litvak, and J. Prochno, On the expectation of operator norms of random matrices , Geometric aspects of functional analysis, Lecture Notes i n Math., vol. 2169, Springer, Cham, 2017, pp. 151–162. MR 3645120

work page 2017