Comparison theorems for the extreme eigenvalues of a random symmetric matrix

Joel A. Tropp

REVIEW 1 major objections 4 minor 2 cited by

Comparison theorems for the extreme eigenvalues of a random symmetric matrix

T0 review · 1 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The top eigenvalue of any sum of independent random symmetric matrices is controlled by the top eigenvalue of a matching Gaussian matrix, plus explicit error terms.

desk verdict Real new comparison theorem and a credible first proof of the lower-distortion half of Nelson-Nguyen, but two public-facing errors need correction before citable. read the letter →

arxiv 2603.04365 v3 pith:CKWEZZ2O submitted 2026-03-04 math.PR cs.NAmath.NAmath.STstat.TH

Joel A. Tropp This is my paper

classification math.PRcs.NAmath.NAmath.STstat.TH MSC 15B5260B20

keywords comparisontheoremrandommatrixextremeeigenvaluesGaussianproxyLindebergmethodtraceexponentialconcentrationsubspaceembedding

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 proves that the largest eigenvalue of a sum of independent random self-adjoint matrices is bounded above by the largest eigenvalue of a Gaussian matrix with matching mean and variance, plus small error terms that decay when no single summand dominates. The error terms depend only on a one-sided fluctuation bound for the summands, the dimension, and two summary statistics of the Gaussian proxy. This transfers the rich toolkit of Gaussian random matrix theory to arbitrary matrix sums, yielding sharper concentration bounds than previous matrix Bernstein and universality results. The method delivers the first complete proof of the injectivity half of a 2013 conjecture on sparse dimension reduction maps, together with new spectral bounds for random regular graphs, random Pauli matrices, and sample covariance matrices.

What carries the argument

The central object is the trace exponential f(t) = Tr e^{A + tH} and Stahl's theorem (the former BMV conjecture), which represents it as the Laplace transform of a positive measure on the interval [λmin(H), λmax(H)]. This representation makes the even derivatives of the trace exponential positive and gives sharp bounds on the Taylor remainder, so that Lindeberg's method — exchanging one random summand for its matching Gaussian at a time — yields a one-sided comparison of trace moment generating functions. The summary statistics of the Gaussian proxy that carry the final bound are the matrix fluctuation φ(Z) = E λmax(Z - E Z) and the weak variance σ_*^2(Z), the maximal variance of the quadrat

What would settle it

Compute for a small explicit example, say a sum of independent Bernoulli rank-one matrices with matching Gaussian proxy, the empirical expectation E λmax(Y) and compare it with the theorem's right-hand side; a single violation for d ≥ 2 would refute Theorem 1.1. More directly, for a fixed matrix A and a sparse random W with λmax(W) ≤ R, evaluate the trace-exponential inequality E Tr e^{A+W} ≤ E Tr e^{A+gX}; a counterexample to this one-matrix step would break the entire chain.

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Extended reading notes

Core claim

The central claim is a stochastic domination result: for an independent sum Y of self-adjoint matrices with two finite moments and a uniform upper bound R_+ on the centered maximum eigenvalues, the Gaussian proxy Z with the same first two moments satisfies E λmax(Y) ≤ E λmax(Z) + sqrt((R_+ φ(Z)/3 + σ_*^2(Z))·2 log d) + R_+ log d / 3, with an analogous tail inequality at level s. Here φ(Z) is the expected fluctuation of the Gaussian maximum and σ_*^2(Z) is the supremum over unit vectors of the variance of quadratic forms of Z. The proof proceeds by a Lindeberg exchange of summands, using Stahl's theorem to show the trace exponential is the Laplace transform of a positive measure and thereby c

Load-bearing premise

The argument rests on Stahl's theorem — that the trace exponential along a line is the Laplace transform of a positive measure; if that positivity failed, the key Taylor-remainder control for the Lindeberg exchange would collapse.

Editorial extensions

If this is right

Any accurate estimate for the maximum eigenvalue of a Gaussian matrix transfers to arbitrary independent matrix sums with matching first two moments, paying only sqrt(log d) error terms.
For Wigner matrices the bound gives E λmax(Y) ≤ 2√d + O(d^{1/4} √log d), with the sharp leading constant 2√d.
For Rademacher covariance matrices, the minimum-eigenvalue comparison reproduces the Bai–Yin first-order limit 1 - 2√ρ when n ≫ d.
For the random Pauli model with N = 2^n, k ≳ n^2 α^{-4} summands suffice to match GUE spectral edges to relative error α, improving prior n^3 and n^4 requirements.
It gives the first complete proof that a SparseStack matrix with column sparsity ζ ≍ α^{-1} log(d/p) and embedding dimension k ≍ α^{-2} d is injective on any d-dimensional subspace with high probability, confirming the lower-distortion half of a 2013 conjecture.

Reading between the lines

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

Editorial inference: the same trace-exponential comparison should extend to matrix martingale difference sequences via Azuma-type exchangeable arguments, as the paper notes but does not develop; this would give Gaussian comparison bounds for adaptive sums.
Editorial inference: the one-sided nature of the bound is likely inherent: a matching two-sided comparison would need control of λmin of summands, not just λmax, and the paper's methods do not address the minimum singular value of rectangular matrices.
Editorial inference: a numerical test on small worst-case sums, such as Bernoulli rank-one summands, could reveal whether the log d factors in the error terms are necessary or an artifact of the proof.
Editorial inference: the Gaussian proxy's weak variance σ_*^2 is often much smaller than the square of the matrix fluctuation, so the theorem is strongest when fluctuations are spread across many directions; constructions with highly localized variance may exhibit the worst-case behavior.

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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 main comparison theorem is a genuine advance: it gives a one-sided Gaussian comparison for the maximum eigenvalue with error terms controlled by R_+ φ(Z) + σ_*^2(Z) and only log d factors, strictly improving the upper bound from Brailovskaya–van Handel in many settings. The proof is transparent, uses Stahl's theorem in a new way via Lindeberg exchange, and I checked the central inequality chain — it holds up. The SparseStack injectivity result (Theorem 3.6) is the first proof of the lower-distortion half of the Nelson–Nguyen conjecture with the conjectured column sparsity ζ = O(α^{-1} log d) and embedding dimension k = O(α^{-2} d). That alone makes the paper significant.

What the paper does well: it states clearly what is new vs. the companion paper and BH24b, gives credible applications across graph theory, quantum information, covariance estimation, and numerical linear algebra, and the applications actually use the one-sided control rather than paying for two-sided assumptions. The companion paper is cited appropriately, and the Stahl-based arguments are parameter-free, not fitted.

Soft spots: two mechanical errors in the manuscript. Proposition 4.11 states the tail bound with the inequality in the wrong direction — it should be a lower bound on the probability that λmax is at least the threshold, not at most. And Conjecture 3.5 says k ≤ Const·α^{-2} log d, but any embedding needs k ≥ d and the paper's own Theorem 3.6 requires k ≳ α^{-2} d. These are genuine typos/statement errors, but neither affects Theorem 1.1 or the proof in Section 4. I also note the paper does not attempt two-sided bounds, so its statements are weaker than BH24b in that respect; the author acknowledges this, so it is not a hidden flaw.

Who should read this: anyone working on matrix concentration, universality, or randomized linear algebra. The paper deserves a serious referee, despite the mechanical errors. My recommendation is to send to peer review, and require the author to correct the direction of Proposition 4.11 and the impossible scaling in Conjecture 3.5 before final acceptance.

Referee Report

1 major / 4 minor

Summary. The paper develops a nonasymptotic comparison theorem for the extreme eigenvalues of a sum of independent random self-adjoint matrices. The main result, Theorem 1.1, states that if Y = Σ W_i with E W_i = 0 and λmax(W_i) ≤ R_+, then the expected maximum eigenvalue of Y is bounded by that of a Gaussian proxy Z ∼ normal(EY, Var Y) plus explicit error terms depending on R_+, the Gaussian fluctuation φ(Z), and the weak variance σ_*^2(Z); an analogous one-sided tail bound is also proved. The proof uses Lindeberg's exchange method together with Stahl's theorem on the trace exponential to compare trace mgfs, then Gaussian concentration to extract eigenvalue bounds. Corollaries cover the minimum eigenvalue, the spectral norm of rectangular sums, and unbounded summands via truncation. Applications are given to random regular graphs, the random Pauli model, sample covariance matrices, and, as the headline application, a SparseStack dimension-reduction map for which the paper claims to prove the lower-distortion part of the Nelson–Nguyen conjecture.

Significance. Theorem 1.1 is a substantial refinement of existing matrix concentration and Gaussian universality results. Its one-sided hypothesis λmax(W_i − EW_i) ≤ R_+ is much weaker than uniform norm control, and the logarithmic factors appear only in the error terms, so the bound is always competitive with the matrix Bernstein inequality and with the Brailovskaya–van Handel comparison results. The proof is self-contained modulo standard external facts, and it is parameter-free: no constants are fitted, and the Gaussian proxy statistics are computed from the input model rather than calibrated to the target. If the central theorem is correct, the applications—especially the SparseStack injectivity result—are notable contributions. The manuscript is clearly written and the proof strategy is transparent, with explicit constants and a clean five-step structure.

major comments (1)

[Section 3.4.3, Conjecture 3.5] As stated, the conjecture is impossible: an embedding from C^n to C^k with k < d cannot be injective on a d-dimensional subspace, yet the conjecture claims k ≤ C α^{-2} log d. This also contradicts Theorem 3.6, which requires k ≥ 16 α^{-2}(d ∨ log(d/p)). Consequently, the abstract's claim that the paper gives 'the first complete proof' of the Nelson–Nguyen conjecture is not accurate as written. The intended conjecture almost certainly has a dimension factor (e.g., k ≤ C α^{-2} d log d), and Theorem 3.6 may well be a stronger statement, but the text must be corrected and the claim rephrased to say precisely which version of the conjecture is being proved.

minor comments (4)

[Section 4.10, Proposition 4.11] The displayed tail bound has the inequality direction reversed: it states P{λmax(Y+Δ) ≤ μ + …} ≤ P{M>R} + d e^{-s}, but the proof and Corollary 1.2 establish an upper tail bound of the form P{λmax(Y+Δ) ≥ μ + …} ≤ P{M>R} + d e^{-s}. This is a clear typographical error and should be fixed.
[Section 2.5.2] In the GUE invariance line, the right-hand side should be U^* X_gue U rather than U^* X_goe U. The current text appears to be a copying error.
[Section 3.1.1] The comparison with [BH24b] cites 'Thm. 3.8' for the random regular graph bound, while elsewhere the relevant result is referred to as Cor. 2.7. Please ensure the reference is consistent.
[Section 3.2.1] In the last paragraph of the proof, the condition 'requiring that 3√k ≤ N' and the earlier theorem assumption k ≤ N^2/9 are consistent, but the wording '3√k' is ambiguous; it would be clearer to state √k ≤ N/3 or k ≤ N^2/9.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity; the central comparison proof is self-contained modulo the external Stahl theorem and auxiliary Gaussian lemmas.

full rationale

Walking the derivation chain from Theorem 1.1, the trace-mgf comparison (Prop 4.6) uses a Gaussian summand X~normal(0, Var W) whose first two moments match W by construction; this is an input statistic, not a fitted output. The scaling factor g is a deterministic function of the upper-bound R, and the inequality follows from Stahl's theorem plus the second-moment matching, not from the target bound. Proposition 4.8 telescopes over independent Lindeberg exchanges, and Proposition 4.9 uses standard Gaussian concentration; Proposition 4.10 only optimizes the resulting mgf bound. The Gaussian proxy statistics phi(Z) and sigma_*^2 appear only on the right-hand side and are computed from the model's moments, so no prediction is statistically forced. Stahl's theorem (Fact 4.3) is an external theorem; it supplies differential properties of the trace exponential rather than a disguised form of the conclusion. The companion-paper citations ([Tro25] for Gaussian monotonicity, Gaussian concentration, and Lemma 5.3 in one application) are auxiliary lemmas with independent content, and they do not uniquely force the main theorem. The skeptic-identified issues—Proposition 4.11's reversed inequality direction and Theorem 3.6 not matching Conjecture 3.5's upper-bound scaling on k—are correctness concerns, not circularity. Overall, no load-bearing circular step was found; the score 1 reflects only the presence of minor, non-load-bearing self-citations in background material.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted constants or ad hoc parameters: all error-term constants are explicit universal numbers. The Gaussian proxy Z is a standard construction, not a new entity. The proof relies on external deep results (Stahl's theorem), standard Gaussian concentration/monotonicity, matrix Laplace transforms, and standard example statistics; several facts are cited from the author's companion paper [Tro25] but are parameter-free and do not build the target result into the input.

assumptions (7)

standard math Stahl's theorem (BMV conjecture): for self-adjoint A,H, Tr e^{A+tH} = ∫ e^{λt} dν(λ) with ν a positive measure (Fact 4.3).
Controls all even/odd derivatives of the trace mgf in Props. 4.4 and 4.6; the core of the Lindeberg comparison.
standard math Gaussian concentration: for any 1-Lipschitz f (e.g. λmax) and Gaussian self-adjoint X, E e^{f(X)−Ef(X)} ≤ e^{σ_*^2(X)/2} (Fact 2.3).
Used in Prop. 4.9 to convert the trace-mgf comparison into a bound involving Eλmax(Z) and σ_*^2.
standard math Gaussian monotonicity in the variance function (Fact 2.2): E f(X) ≤ E f(X') for convex f when Var[X] ≤ Var[X'].
Used in the truncation comparison (4.10)-(4.11) and to justify replacing the proxy by a more variable Gaussian.
standard math Matrix Laplace transform method (Fact 4.2): Eλmax(Y) ≤ inf_θ θ^{−1} log E Tr e^{θY}, with an analogous tail bound.
Bridges the trace-mgf bound to the final eigenvalue expectation and tail statements.
standard math Moment-tensor equivalence: same expectation plus same variance function implies E[W⊗W] = E[X⊗X] (Fact 2.1).
Ensures the second-derivative terms in Prop. 4.6 cancel after exchanging W for the Gaussian.
domain assumption Independent summands with two finite moments and one-sided bound λmax(W_i−EW_i) ≤ R_+ (Eq. 1.5).
The hypothesis of Theorem 1.1; the proof uses it only on the upper side, enabling the minimum-eigenvalue corollary.
standard math Standard Gaussian fluctuation bounds used in examples: Chevet's theorem, Slepian's lemma, GOE/GUE estimates [AS17], and the Bai-Yin law.
Computes or calibrates φ(Z), σ_*^2(Z), and Eλmax for the Wigner, covariance, Pauli, and graph examples.

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Cite this review

Pith. "Pith review of Comparison theorems for the extreme eigenvalues of a random symmetric matrix." pith.science (2026). https://pith.science/paper/CKWEZZ2O

@misc{pith2026260304365,
  author       = {Pith},
  title        = {Pith review of: Comparison theorems for the extreme eigenvalues of a random symmetric matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKWEZZ2O}},
  note         = {Machine review of arXiv:2603.04365}
}

read the original abstract

This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix whose statistics match the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm; the proof strategy also extends to matrix martingale sequences. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Well-invertible column subsets of sparse matrices are rare
math.PR 2026-07 accept novelty 7.5 of 10

Under mild sparsity and overlap assumptions, almost every proportional-size column subset of a sparse matrix has smallest singular value o(1), so constant-sparsity SparseStack maps are not (Ω(k), Ω(1))-OSI.
Well-invertible column subsets of sparse matrices are rare
math.PR 2026-07 accept novelty 7.0 of 10

Constant-row-sparsity matrices cannot be oblivious subspace injections for proportional subspace dimension, because well-invertible column subsets of sparse matrices are overwhelmingly rare.

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

A generalization of the Lindeberg principle

Automata, languages and programming. 2004, pages 3–15.doi:10.1016/S0304-3975(03)00400-6. [Cha06] S. Chatterjee. “A generalization of the Lindeberg principle”.Ann. Probab.34.6 (2006), pages 2061–2076.doi: 10.1214/009117906000000575. [Che+24] C.-F. Chen et al. “Sparse Random Hamiltonians Are Quantumly Easy”.Phys. Rev. X14 (1 2024), page 011014. doi:10.1103/...

doi:10.1016/s0304-3975(03)00400-6 2004
[5]

Stahl’s theorem (aka BMV conjecture): insights and intuition on its proof

arXiv:2508.14234 [cs.DS]. [Cli16] F. Clivaz. “Stahl’s theorem (aka BMV conjecture): insights and intuition on its proof”. In:Spectral theory and mathematical physics. Volume

arXiv
[13]

User-friendly tail bounds for sums of random matrices

[Tro12] J. A. Tropp. “User-friendly tail bounds for sums of random matrices”.Found. Comput. Math.12.4 (2012), pages 389–434.doi:10.1007/s10208-011-9099-z. [Tro15] J. A. Tropp. “An introduction to matrix concentration inequalities”.Found. Trends Machine Learning8.1-2 (2015), pages 1–230.doi:10.1561/2200000048. [Tro25] J. A. Tropp. “Comparison theorems for ...

doi:10.1007/s10208-011-9099-z 2012
[14]

Characteristic vectors of bordered matrices with infinite dimensions

arXiv:2501.16578 [math.PR]. [Ver25] R. Vershynin.High-dimensional probability: An introduction with applications in data science. 2nd. Cambridge Univ. Press, 2025.doi:10.1017/9781108231596. [Wig55] E. P. Wigner. “Characteristic vectors of bordered matrices with infinite dimensions”.Ann. of Math. (2)62 (1955), pages 548–564.doi:10.2307/1970079

arXiv 2025
[26]

Random matrices: universality of ESDs and the circular law

IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2019, pages 461–498. [TV10] T. Tao and V. Vu. “Random matrices: universality of ESDs and the circular law”.Ann. Probab.38.5 (2010). With an appendix by Manjunath Krishnapur, pages 2023–2065.doi:10.1214/10-AOP534. [TV11] T. Tao and V. Vu. “Random matrices: universality of local eigenvalue statistic...

doi:10.1214/10-aop534 2019
[29]

OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings

Cambridge Univ. Press, 2020, pages 403–572. [NN13] J. Nelson and H. L. Nguyen. “OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings”. In:FOCS’13: Proc. 2013 IEEE 54th Ann. Symp. Foundations of Computer Science. IEEE Computer Soc., Los Alamitos, CA, 2013, pages 117–126.doi:10.1109/FOCS.2013.21. [NN14] J. Nelson and H. L. Nguye...

work page doi:10.1109/focs.2013.21 2020
[254]

Theory Adv

Oper. Theory Adv. Appl. Birkhäuser/Springer, [Cham], 2016, pages 107–117. doi:10.1007/978-3-319-29992-1\_6. [Dud02] R. M. Dudley.Real analysis and probability. Revised reprint of the 1989 original. Cambridge Univ. Press,

work page doi:10.1007/978-3-319-29992-1 2016
[2002]

Herbert Stahl’s proof of the BMV conjecture

doi:10.1017/CBO9780511755347. [Erë15] A. È. Erëmenko. “Herbert Stahl’s proof of the BMV conjecture”.Mat. Sb.206.1 (2015), pages 97–102.doi: 10.4213/sm8294. 32 REFERENCES [Fri08] J. Friedman. “A proof of Alon’s second eigenvalue conjecture and related problems”.Mem. Amer. Math. Soc. 195.910 (2008), pages viii+100.doi:10.1090/memo/0910. [Hei25] O. Heinävaar...

doi:10.1017/cbo9780511755347 2015

Show all 14 references

[2019]

Gaussian measures of dilatations of convex symmetric sets

[LO99] R. Latała and K. Oleszkiewicz. “Gaussian measures of dilatations of convex symmetric sets”.Ann. Probab.27.4 (1999), pages 1922–1938.doi:10.1214/aop/1022677554. [LT91] M. Ledoux and M. Talagrand.Probability in Banach spaces. Isoperimetry and processes. Springer-Verlag, B...

1999
[2023]

Applications of the Lindeberg principle in communications and statistical learning

[KM11] S. B. Korada and A. Montanari. “Applications of the Lindeberg principle in communications and statistical learning”.IEEE Trans. Inform. Theory57.4 (2011), pages 2440–2450.doi:10.1109/TIT.2011.2112231. [Kow19] E. Kowalski.An introduction to expander graphs. Société Mathé...

2011
[2024]

Universality and Sharp Matrix Concentration Inequalities

[BH24b] T. Brailovskaya and R. van Handel. “Universality and Sharp Matrix Concentration Inequalities”.Geom. Funct. Anal.34.6 (2024), pages 1734–1838.doi:10.1007/s00039-024-00692-9. [Cam+25] C. Camaño et al. “Faster linear algebra algorithms with structured random matrices”. Av...

2024 doi
[2025]

Finding frequent items in data streams

arXiv:2508.21189. [CCFC04] M. Charikar, K. Chen, and M. Farach-Colton. “Finding frequent items in data streams”. In: volume

arXiv
[2026]

Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix

arXiv: 2602.05394 [math.NA]. [AS17] G. Aubrun and S. J. Szarek.Alice and Bob meet Banach: The interface of asymptotic geometric analysis and quantum information theory. AMS, 2017.doi:10.1090/surv/223. [BY93] Z. D. Bai and Y. Q. Yin. “Limit of the smallest eigenvalue of a large...

2017
[8572]

The lower tail of random quadratic forms with applications to ordinary least squares

Lecture Notes in Comput. Sci. Springer, 2014, pages 883–894.doi: 10.1007/978-3-662-43948-7_73. [Oli16] R. I. Oliveira. “The lower tail of random quadratic forms with applications to ordinary least squares”.Probab. Theory Related Fields166.3-4 (2016), pages 1175–1194.doi:10.100...

2014 doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

A generalization of the Lindeberg principle

Automata, languages and programming. 2004, pages 3–15.doi:10.1016/S0304-3975(03)00400-6. [Cha06] S. Chatterjee. “A generalization of the Lindeberg principle”.Ann. Probab.34.6 (2006), pages 2061–2076.doi: 10.1214/009117906000000575. [Che+24] C.-F. Chen et al. “Sparse Random Hamiltonians Are Quantumly Easy”.Phys. Rev. X14 (1 2024), page 011014. doi:10.1103/...

doi:10.1016/s0304-3975(03)00400-6 2004

[2] [5]

Stahl’s theorem (aka BMV conjecture): insights and intuition on its proof

arXiv:2508.14234 [cs.DS]. [Cli16] F. Clivaz. “Stahl’s theorem (aka BMV conjecture): insights and intuition on its proof”. In:Spectral theory and mathematical physics. Volume

arXiv

[3] [13]

User-friendly tail bounds for sums of random matrices

[Tro12] J. A. Tropp. “User-friendly tail bounds for sums of random matrices”.Found. Comput. Math.12.4 (2012), pages 389–434.doi:10.1007/s10208-011-9099-z. [Tro15] J. A. Tropp. “An introduction to matrix concentration inequalities”.Found. Trends Machine Learning8.1-2 (2015), pages 1–230.doi:10.1561/2200000048. [Tro25] J. A. Tropp. “Comparison theorems for ...

doi:10.1007/s10208-011-9099-z 2012

[4] [14]

Characteristic vectors of bordered matrices with infinite dimensions

arXiv:2501.16578 [math.PR]. [Ver25] R. Vershynin.High-dimensional probability: An introduction with applications in data science. 2nd. Cambridge Univ. Press, 2025.doi:10.1017/9781108231596. [Wig55] E. P. Wigner. “Characteristic vectors of bordered matrices with infinite dimensions”.Ann. of Math. (2)62 (1955), pages 548–564.doi:10.2307/1970079

arXiv 2025

[5] [26]

Random matrices: universality of ESDs and the circular law

IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2019, pages 461–498. [TV10] T. Tao and V. Vu. “Random matrices: universality of ESDs and the circular law”.Ann. Probab.38.5 (2010). With an appendix by Manjunath Krishnapur, pages 2023–2065.doi:10.1214/10-AOP534. [TV11] T. Tao and V. Vu. “Random matrices: universality of local eigenvalue statistic...

doi:10.1214/10-aop534 2019

[6] [29]

OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings

Cambridge Univ. Press, 2020, pages 403–572. [NN13] J. Nelson and H. L. Nguyen. “OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings”. In:FOCS’13: Proc. 2013 IEEE 54th Ann. Symp. Foundations of Computer Science. IEEE Computer Soc., Los Alamitos, CA, 2013, pages 117–126.doi:10.1109/FOCS.2013.21. [NN14] J. Nelson and H. L. Nguye...

work page doi:10.1109/focs.2013.21 2020

[7] [254]

Theory Adv

Oper. Theory Adv. Appl. Birkhäuser/Springer, [Cham], 2016, pages 107–117. doi:10.1007/978-3-319-29992-1\_6. [Dud02] R. M. Dudley.Real analysis and probability. Revised reprint of the 1989 original. Cambridge Univ. Press,

work page doi:10.1007/978-3-319-29992-1 2016

[8] [2002]

Herbert Stahl’s proof of the BMV conjecture

doi:10.1017/CBO9780511755347. [Erë15] A. È. Erëmenko. “Herbert Stahl’s proof of the BMV conjecture”.Mat. Sb.206.1 (2015), pages 97–102.doi: 10.4213/sm8294. 32 REFERENCES [Fri08] J. Friedman. “A proof of Alon’s second eigenvalue conjecture and related problems”.Mem. Amer. Math. Soc. 195.910 (2008), pages viii+100.doi:10.1090/memo/0910. [Hei25] O. Heinävaar...

doi:10.1017/cbo9780511755347 2015