Statistics of Min-max Normalized Eigenvalues in Random Matrices

Hyakka Nakada; Shu Tanaka

REVIEW 4 major objections 5 minor 39 references

Statistics of Min-max Normalized Eigenvalues in Random Matrices

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For a Gaussian random matrix, the distribution of min-max normalized eigenvalues and the low-rank coupling error depend, in the large-N limit, only on the ratio J1/J0, and the paper derives explicit closed-form formulas for both.

desk verdict The asymptotic scaling/plateau results hold up; the finite-N coupling-error formula (16) is wrong for α∈(r,1) due to a smeared largest-eigenvalue atom — worth peer review after revision. read the letter →

arxiv 2512.15427 v1 pith:LX2XWUV5 submitted 2025-12-17 cs.LG cond-mat.stat-mechmath.STstat.TH

Hyakka Nakada , Shu Tanaka This is my paper · ORCID

classification cs.LGcond-mat.stat-mechmath.STstat.TH MSC 60B2015B52

keywords randommatrixtheorymin-maxnormalizationeigenvaluedistributionWignersemicirclelawGaussianfactorizationcouplingerrorscaling

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

Most data-processing pipelines normalize inputs before using them, and eigenvalues are no exception. This paper studies the eigenvalues of a Gaussian random matrix after min-max scaling, so that the smallest eigenvalue becomes 0 and the largest becomes 1. It claims that, once the matrix is large, the whole distribution of these normalized eigenvalues and the residual error of a low-rank factorization of Q − λ_N I depend only on the ratio of the two coupling parameters J1/J0, not on their absolute values. The paper derives explicit formulas — a cumulative distribution with a semicircle-shaped body and a plateau, and a coupling error that saturates at the constant 5 once the truncation threshold passes the point r — and verifies them numerically for matrices up to 500×500. If correct, this gives a closed-form way to anticipate the truncation error from the ratio alone, without diagonalizing the matrix.

What carries the argument

The central object is the approximate effective cumulative distribution P(λ̂<x) given in Eqs. (4)–(5), imported from the authors' earlier study. It is built on replacing the extreme eigenvalues by their Wigner-semicircle and largest-eigenvalue expectations (3), which yields a deterministic threshold r = 4(J1/J0)/(1+J1/J0)² for the normalized second-largest eigenvalue. The derivation then uses a continuum approximation to turn sums over normalized eigenvalues into integrals against this distribution, giving both the scaling-law CDFs and the expected coupling error. All final formulas depend on the matrix only through the single ratio J1/J0.

What would settle it

Measure the empirical cumulative distribution of min-max normalized eigenvalues for a single Gaussian matrix of size N≈100 with J1/J0=0.3, and look near x=1: Eq. (4) predicts a linear rise with slope 1/((1−r)N) throughout (r,1), whereas the true distribution should be flat until a jump of size 1/N at x=1; whichever shape appears decides whether the continuum CDF is reliable.

Watch

Extended reading notes

Core claim

Starting from the approximate cumulative distribution of normalized eigenvalues from their previous study, the paper derives a large-N scaling law: for J1 ≤ J0 the CDF converges to a semicircle-type integral capped at r = 4(J1/J0)/(1+J1/J0)², and for J1 > J0 to a universal semicircle integral. From the same distribution it evaluates the expected coupling error of factoring Q − λ_N I as VV^T, obtaining a formula that plateaus at 5 once the threshold α passes r. Both results depend on the matrix only through J1/J0 — a property absent for unnormalized eigenvalues.

Load-bearing premise

The results rest on the approximate cumulative distribution (4)–(5) taken from the authors' prior work, which spreads the probability mass of the largest normalized eigenvalue uniformly over (r,1) instead of leaving a single atom at 1; if that distribution is inaccurate, the finite-N error formulas fail, although the asymptotic plateau may still hold.

Editorial extensions

If this is right

Two Gaussian matrices with different absolute scales but the same J1/J0 will show identical normalized eigenvalue CDFs and identical normalized coupling errors; a single parameter captures the spectrum.
The expected coupling error per N J1² saturates at 5 for truncation thresholds α above r, so the entire truncation error is carried by the normalized eigenvalues below r.
For J1 > J0 the normalized spectrum and error become universal, independent of the ratio's value, which simplifies the noise-dominated regime.
The finite-N formulas include explicit 1/N corrections, and the numerical results show convergence from N=100 upward, so the asymptotic predictions are practically usable.
The closed-form expressions allow a practitioner to choose a rank or threshold α that achieves a desired coupling error without performing an eigendecomposition.

Reading between the lines

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

Because Eq. (4) spreads the largest eigenvalue's mass uniformly over (r,1), the α-dependent term in the finite-N formula (16) for α>r is likely an artifact of that approximation; a CDF that places an atom at x=1 would produce a flat plateau already at finite N.
If the ratio-only behavior extends beyond Gaussian entries — as semicircle-law universality suggests — the same normalized CDF and error formulas could apply to other random-matrix ensembles, making the result a general low-rank approximation tool.
In factorization-machine and black-box optimization settings where the coupling error bounds the regression error, this implies the achievable accuracy is governed by J1/J0 alone, so rank or threshold selection can be guided by this single parameter.

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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

Two things you should know. First, the paper's headline asymptotic results — the scaling law (9),(10) and the large-N coupling-error plateau (18),(19) — are correct and well supported by the numerics. Second, the finite-N coupling-error formula (16) is not correct for α in (r,1): it inherits a spurious α-dependence from the way Eq. (4) smears the largest-eigenvalue atom. The stress-test note is right, and the flaw is locatable, not a matter of taste.

What the paper does well: it takes the approximate CDF from the authors' prior work [13] and straightforwardly derives the consequences. The scaling-law observation (distribution depends only on J1/J0) is genuinely useful and the numerical verification (Figs. 1, 2) is clean. The algebra from (4),(5) to the integration formulas is internally consistent, and the asymptotic plateau at 5 is visible in the experiments.

Soft spots, in order. The main one is Eq. (16). In (4), for x>r the CDF grows linearly from 1−1/N to 1, i.e. the 1/N mass of λ̂1=1 is spread uniformly over [r,1]. Plugging that into (15) gives the α^3 term in (16). But physically λ̂1=1, so for any α<1 the set {λ̂_i<α} excludes λ1 and the sum is constant on (r,1). The α-dependence in (16) is an artifact; the plateau (18) survives because the smeared mass is O(1/N). So (16) should not be presented as a finite-N prediction. There is also a minor inconsistency: Eq. (3) uses λ1≈Nμ+σ^2/μ, while Eq. (5) (J1>J0) effectively uses λ1−λN≈4√Nσ; those two approximations don't connect, and the paper doesn't clarify which is in force. Also, the Fig. 3 caption says the theory line is from (16)/(17) while the text says (18)/(19); for J1=0.1,0.3 those differ at finite N, so the reader can't tell which was actually plotted.

Bottom line: the asymptotic results are worth having, the finite-N claim needs a correction. The paper deserves a serious referee, but I'd ask the authors to revise (16) or at least restrict it to the asymptotic regime and fix the inconsistencies. Practitioners using low-rank factorization of noisy matrices should use the plateau formulas, not (16), for finite N. I'd bring it to a reading group as a lesson in finite-size artifacts in continuum approximations, but it's not a must-read.

Referee Report

4 major / 5 minor

Summary. The paper studies the min-max normalized eigenvalues λ̂ = (λ - λ_N)/(λ_1 - λ_N) of a random matrix Q whose off-diagonal entries are i.i.d. N(μ, σ²) and whose diagonal entries are N(μ, 2σ²). Starting from an effective cumulative distribution function CDF proposed in the authors' earlier work [13], the paper derives (i) a scaling law stating that in the large-N limit the normalized empirical CDF depends only on the ratio J₁/J₀ = σ√N / μN, and (ii) closed-form expressions for the expected 'coupling error' when Q - λ_N I is approximated by a rank-k factorization. The main results are the CDF formulas (9), (10), the finite-N coupling-error formulas (16), (17), and their large-N limits (18), (19). Numerical experiments with N up to 500 and various J₁/J₀ are presented to support the theory.

Significance. If correct, the asymptotic formulas (9), (10), (18), (19) provide simple, parameter-light predictions for the spectrum and truncation error of a random matrix after min-max normalization, which could be useful for practitioners in matrix factorization and related ML models. The derivations from (4), (5) to (16)–(19) are explicit and the numerical experiments are reproducible. The main weakness is that the finite-N coupling-error formula (16) contains a qualitative error in the α > r regime, and the paper has an inconsistency between the text and the Fig. 3 caption about which formula is plotted. These issues affect the central finite-N claims, while the large-N plateau (18) appears sound.

major comments (4)

[§3, Eq. (16)] The α>r branch of Eq. (16) has a spurious α-dependence. By definition, the largest min-max normalized eigenvalue is exactly λ̂₁ = 1. Therefore, for any α with r < α < 1, the set {i : λ̂_i < α} is exactly {2,...,N}, and the sum Σ_{λ̂_i<α} λ̂_i² cannot depend on α. Equation (4), however, spreads the 1/N mass of the largest eigenvalue uniformly over [r,1]. This smearing directly produces the term 16(α³-r³)/(3N(1-r)r²) in Eq. (16), which is an artifact. The correct finite-N expression in this regime should be constant, e.g. 5(N-2)/N under the same approximation. The large-N limit (18) survives because the spurious term is O(1/N), but Eq. (16) cannot be presented as the finite-N analytical result.
[§3, Fig. 3 caption vs. text] The Fig. 3 caption states the black theoretical line is computed from Eq. (16) or (17), while the text says Eq. (18) or (19) is used. The plateau at value 5 for α>r, which the text explicitly associates with r, is consistent only with the asymptotic formula (18), not with (16). This inconsistency obscures the defect in Eq. (16). The authors must correct the caption/text mismatch and clearly state which formula is shown. If the data are compared with the asymptotic formulas (18)/(19), then finite-N formula (16) should not be presented as the verified finite-N prediction.
[§2, Eq. (3) and Eqs. (4)–(5)] Equation (3) gives λ₁ ≈ Nμ + σ²/μ and λ_N ≈ -2√Nσ, and the text implies these approximations lead to both (4) and (5). However, for σ > √Nμ (i.e. J₁ > J₀) the largest eigenvalue does not separate from the bulk; the correct approximation is λ₁ ≈ 2√Nσ, so that λ₁ - λ_N ≈ 4√Nσ. Equation (5) and the subsequent derivation of (17) implicitly use this latter value, not the λ₁ of Eq. (3). The paper should specify the regime of validity of Eq. (3) and explain how Eq. (5) is obtained. As written, the derivation is inconsistent and not self-contained.
[§2, Eq. (4)] The effective CDF (4) is imported from reference [13] without proof or error bounds. In particular, the treatment of the largest normalized eigenvalue (spread uniformly over [r,1] rather than an atom at x=1) is a modeling assumption that directly controls the finite-N coupling-error formula (16). Since this is a load-bearing premise for the finite-N claims, the authors should either provide a derivation or error analysis of (4), or explicitly restrict the claims to the large-N limit where the atom's location is immaterial.

minor comments (5)

[Title] The title has a typo: 'Mat rices' should be 'Matrices'.
[Fig. 2 caption] 'varing N' should be 'varying N'.
[Fig. 3 caption] The phrase 'with fixing J0 = 1' is grammatically awkward; suggest 'with J0 fixed to 1'.
[Various] There are several minor typos and grammatical issues (e.g., 'suﬃciently', 'theoretically and experimentally veriﬁed'). A careful proofread is recommended.
[References] Reference [13] is an arXiv preprint; if a published version exists, it should be cited. Also, reference [35] may be updated if a journal version is available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: scaling-law and coupling-error formulas are derived from a disclosed prior effective CDF, not fitted to the target data; numerical experiments provide external checks.

full rationale

The paper is explicit about its starting point: Eqs. (4) and (5), the approximate CDF of min-max normalized eigenvalues, are taken from the authors' previous work [13]. The new results are obtained from this input by taking the N→∞ limit to get the scaling-law CDFs (9)–(10), and by inserting the CDF into the expectation integral (15) to get the coupling-error formulas (16)–(19). No parameter in the output is fitted to the simulated eigenvalues in this paper: the ratio r is set by Eq. (8) from J0 and J1, and the prefactor 16J1²/r² comes from the deterministic approximation of λ1−λN in Eq. (14). The numerical experiments in Figs. 1–4 are external checks against sampled Gaussian matrices, so the self-citation [13] is not the only support for the model. There is a validity concern in the finite-N formula (16): the x>r branch of the input CDF smears the largest-eigenvalue atom over [r,1], producing an α-dependence for α∈(r,1) that the true empirical sum Σ_{λ̂i<α}λ̂i² does not have; this is a modeling/approximation artifact rather than a circular reduction, and it disappears in the asymptotic plateau (18). The caption of Fig. 3 and the text also disagree about whether (16)/(17) or (18)/(19) is plotted. These are correctness/consistency issues, not cases where a prediction is equivalent to an input by construction.

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

The central results rest on the approximate CDF (4),(5) from the authors' prior paper [13], which in turn relies on Wigner's semicircle law and largest-eigenvalue asymptotics. No new free parameters are fitted to data in this paper; J0, J1, N enter as model inputs. The main unstated assumption is the smearing of the largest normalized eigenvalue in Eq. (4).

assumptions (6)

standard math Wigner semicircle law for bulk eigenvalues of the Gaussian orthogonal ensemble
Used in Eq. (3) for λ2≈2√Nσ and λN≈−2√Nσ, and in the bulk CDF components of Eqs. (4),(5).
domain assumption Largest eigenvalue of the spiked Wigner matrix is approximated by Nμ+σ^2/μ
Eq. (3); this formula is valid only when the spike is strong (J1<J0), but the paper states it without this caveat, and the J1>J0 branch of Eq. (5) effectively uses λ1−λN≈4√Nσ.
domain assumption The effective CDF (4),(5) from ref [13] is accepted as given
The present paper derives all subsequent results from this CDF and does not re-derive or bound its error.
ad hoc to paper In Eq. (4), the 1/N mass of the largest eigenvalue is spread uniformly over [r,1]
This smearing replaces the atom at x=1; it enters the finite-N coupling-error formula (16) for α>r and is not physically motivated.
domain assumption Continuum approximation replaces the eigenvalue sum with N∫x²dP in Eq. (15)
Assumes smooth distribution for large N; introduces O(1/N) corrections that are not quantified.
domain assumption Deterministic replacement of λ1−λN by E[λ1]−E[λN]
Eq. (14), used to convert sums over absolute eigenvalues to sums over normalized eigenvalues; ignores fluctuations of the spectral edges.

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Cite this review

Pith. "Pith review of Statistics of Min-max Normalized Eigenvalues in Random Matrices." pith.science (2026). https://pith.science/paper/LX2XWUV5

@misc{pith2026251215427,
  author       = {Pith},
  title        = {Pith review of: Statistics of Min-max Normalized Eigenvalues in Random Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LX2XWUV5}},
  note         = {Machine review of arXiv:2512.15427}
}

read the original abstract

Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus, this study investigates the statistical properties of min-max normalized eigenvalues in random matrices. Previously, the effective distribution for such normalized eigenvalues has been proposed. In this study, we apply it to evaluate a scaling law of the cumulative distribution. Furthermore, we derive the residual error that arises during matrix factorization of random matrices. We conducted numerical experiments to verify these theoretical predictions.

Figures

Figures reproduced from arXiv: 2512.15427 by the authors.

**Figure 1.** (Color online) Plots of the cumulative distribution of min-max normalized eigenvalues with varying J1/J0. The black lines show our theoretical line computed from Eq. (4) or (5), for J1/J0 = 0.1, 0.3, 10.0 respectively. The green, blue, and red lines are experimental plots for J0 = 0.1, 1.0, 10.0, respectively. These lines were obtained with N = 100. distribution (4), (5) and (6), several statistics are derived and… view at source ↗

**Figure 3.** (Color online) Plots of coupling errors as a function of ratio α. The black line shows our theoretical prediction computed from Eq. (16) or (17). The yellow, green, blue and red lines represent the experimental results for ∆α. These lines shall guide the eye. The colored areas indicate the standard deviations. With fixing J0 = 1, the results of J1 = 0.1 (left), 0.3 (middle), and 10.0 (right) are shown [PITH_FULL_IM… view at source ↗

**Figure 4.** (Color online) Cropped and enlarged view of [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [13]

Y . Seki, H. Nakada, and S. Tanaka: arXiv preprint arXiv:2 410.12747 (2024)

2024

[2] [1]

Perry, S

R. Perry, S. Panigrahi, J. Bien, and D. Witten: Journal of t he American Statistical Association (2025) 1

2025

[3] [2]

M. R. Zirnbauer: Journal of Mathematical Physics 37 (1996) 4986

1996

[4] [3]

J. P . Keating and N. C. Snaith: Communications in Mathemat ical Physics 214 (2000) 57

2000

[5] [4]

E. P . Wigner: SIAM review 9 (1967) 1

1967

[6] [5]

T. A. Brody, J. Flores, J. B. French, P . A. Mello, A. Pandey, and S. S. Wong: Reviews of Modern Physics 53 (1981) 385

1981

[7] [6]

C. W. Beenakker: Reviews of modern physics 69 (1997) 731

1997

[8] [7]

A. M. Tulino, S. V erd´ u, et al.: Foundations and Trends® in Communi- cations and Information Theory 1 (2004) 1

2004