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A Wishart matrix is close in total variation to a shifted GOE once k is larger than n cubed, with an explicit square-root rate.

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2026-07-12 13:53 UTC pith:MTMDWSLP

load-bearing objection Clean non-asymptotic TV rate between Wishart and shifted GOE; useful lemma already cited, no real holes.

arxiv 2606.16018 v2 pith:MTMDWSLP submitted 2026-06-14 math.PR

A non-asymptotic bound on the TV distance between a Wishart matrix and an appropriately scaled GOE matrix

Raphael Arkady Meyer This is my paper
classification math.PR MSC 60B2062H1015B52
keywords Wishart matrixGOEtotal variationnon-asymptotic boundPinsker inequalityrandom matrix theoryKL divergence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This note supplies a quantitative, finite-n version of a known asymptotic closeness result between two classical random-matrix ensembles. It shows that a Wishart matrix with k degrees of freedom and an affine transformation of a GOE matrix of the same size become indistinguishable in total variation as soon as k is a constant multiple of n cubed; the total-variation distance is then at most a constant times the square root of n cubed over k. The argument follows the earlier asymptotic proof but replaces the final limiting step by an explicit KL-to-TV conversion via Pinsker’s inequality and careful control of a high-probability spectral-norm event. The resulting bound is already strong enough to be invoked as a black-box comparison tool in later algorithmic and spectral-density work.

Core claim

If G is drawn from the Gaussian orthogonal ensemble of size n and W is a Wishart matrix with the same dimension and k degrees of freedom, then the total-variation distance between W and the shifted matrix √k G + k I is at most on the order of √(n³/k) whenever k is larger than a constant multiple of n³.

What carries the argument

An explicit upper bound on the log-likelihood ratio between the shifted-GOE density and the Wishart density (Theorem 3.1), obtained by Stirling approximation of the Wishart normalizing constant followed by a degree-3 Taylor expansion of the resulting scalar function β whose remainder is controlled on the spectral-norm event {‖G‖₂ ≤ ½√k}.

Load-bearing premise

The argument relies on a high-probability bound for the spectral norm of a GOE matrix that keeps every eigenvalue of the shifted matrix safely above k/2; if the hidden constant in that bound is larger than assumed, the threshold k ≳ n³ must be increased.

What would settle it

Compute the exact total-variation distance (or a tight Monte-Carlo estimate of it) for moderate n and for a sequence of k that straddle the claimed threshold n³; if the observed distance remains order-1 for k ≫ n³, or drops below the claimed √(n³/k) rate for k ≪ n³, the bound is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. This note proves a non-asymptotic total-variation bound between a Wishart matrix W ~ Wishart(n, k) and the affine image bG = √k G + k I of a GOE matrix: if k ≳ n^{3} then TV(W, bG) ≲ √(n^{3}/k). The argument follows the density-ratio strategy of Bubeck–Ding–Eldan–Rácz, replacing their asymptotic approximations by an explicit Stirling bound on the Wishart normalizing constant, a degree-3 Taylor expansion of the resulting log-ratio function eta with remainder controlled on the high-probability event {||G||_{2} ≤ (1/2)√k}, conditional GOE moment estimates, and Pinsker’s inequality applied to the conditioned law of bG.

Significance. The bound supplies a clean quantitative version of a previously asymptotic comparison that is already invoked in concurrent work on matrix-vector query complexity and spectral density estimation. The removal of logarithmic factors via KL divergence + Pinsker is a modest but useful technical improvement, and the proof is entirely elementary once the classical Wishart/GOE densities and the imported spectral-norm and moment bounds are granted. The result is therefore of immediate utility as a black-box lemma.

minor comments (5)
  1. Abstract attributes the original result to “Rácz and Richey,” while the introduction, Theorem 1.3 discussion, and reference list correctly identify Bubeck–Ding–Eldan–Rácz (2016, Thm. 4). Correct the attribution for consistency.
  2. §3.1: the event A is written ||G||_{2} ≤ (1/2)√k and claimed to hold with probability 1-e^{-k} via Theorem 2.7. Because the ≲ of Vershynin hides a universal constant C that is typically larger than 1/2, the numerical threshold 1/2 is formally insufficient. Replacing 1/2 by a sufficiently small absolute constant c and absorbing the resulting factor into the hypothesis k ≳ n^{3} repairs the claim without changing the rate; a one-sentence remark would make the dependence transparent.
  3. Lemma 2.3: the proof introduces “p ≤ δ” without having defined δ inside the lemma statement; the only hypothesis is that the event holds with probability at least 1/2. Replacing the undefined δ by 1/2 removes the ambiguity.
  4. §3.2, Stirling step: the error term arising from log Γ(x) = h(x) + O(1/x) is summed to O(n/k) and then absorbed into the final O(n^{3}/k). A parenthetical note tracking this contribution would help a reader verify that no larger error is hidden.
  5. Throughout: the symbols ≲ and ≳ are used for absolute constants. A single sentence in §2 stating that all implicit constants are universal (independent of n and k) would make the notation self-contained.

Circularity Check

0 steps flagged

No circularity: classical densities + standard inequalities yield a non-asymptotic TV bound without feeding the target back into the hypotheses.

full rationale

The derivation is self-contained. It starts from the classical Wishart and GOE densities (Facts 2.5–2.6), applies Stirling’s approximation to the Gamma factors, forms the log-likelihood ratio α(A), expands it via a degree-3 Taylor remainder for the scalar function β, and controls the remainder under the spectral-norm event A = {‖G‖₂ ≤ ½√k}. Conditioning is justified by the elementary non-negative-expectation lemma and rotational invariance; the resulting conditional moments are bounded by Tao’s unconditional moments. Pinsker’s inequality then converts the KL bound into the claimed TV rate √(n³/k). The only self-citations are to the asymptotic predecessor being strengthened and to two later papers that use the present result; none of them is load-bearing for the proof. No fitted parameter is renamed a prediction, no uniqueness theorem is imported from the authors, and the target TV distance never re-enters the hypotheses. Score 0 is therefore appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

Pure-math note; the only external inputs are classical density formulae, Pinsker’s inequality, Stirling’s approximation, Vershynin’s spectral-norm tail, and Tao’s GOE moment bounds. No fitted constants or postulated entities appear.

axioms (5)
  • standard math Pinsker’s inequality: TV(P,Q) ≤ √(½ D_KL(P∥Q))
    Invoked in §3.1 to convert the KL bound into the final TV claim.
  • standard math Stirling approximation log Γ(x) ≤ h(x) + O(1/x) with the explicit h given in §3.2
    Used to control the product of Gamma factors in the Wishart density; the summed O(n³/k) error is absorbed into the final bound.
  • standard math Vershynin Thm. 4.4.3: ‖G‖₂ ≲ √n + √log(1/δ) with probability 1−δ for G ~ GOE(n)
    Supplies the high-probability spectral event A that keeps all eigenvalues of the shifted matrix above k/2.
  • standard math Tao (Topics in RMT, p. 139): E[tr(G)]=0, E[tr(G²)]≲n², E[tr(G³)]=0, E[tr(G⁴)]≲n³
    Used after conditioning on A (Lemma 2.9) to evaluate the expectation of the log-density ratio.
  • standard math Classical closed-form densities of Wishart(n,k) and GOE(n) with respect to Lebesgue measure on symmetric matrices
    Starting point of the log-ratio calculation in §3.2.

pith-pipeline@v1.1.0-grok45 · 11059 in / 2640 out tokens · 35017 ms · 2026-07-12T13:53:50.525017+00:00 · methodology

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read the original abstract

In this note, we prove a non-asymptotic version of a theorem by R\'acz and Richey, showing that a Wishart matrix is close in total variation to an affine transformation of a GOE matrix. The proof mirrors a proof in a paper by Bubeck, Ding, Eldan, and R\'acz, with some changes made to make it non-asymptotic.

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Reference graph

Works this paper leans on

6 extracted references · 1 linked inside Pith

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    Micha Derezi \'n ski, Ethan N Epperly, and Raphael A Meyer. The matrix-vector complexity of ax= b . arXiv preprint arXiv:2602.04842 , 2026

    arXiv 2026

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    Spectral density estimation for normal matrices

    Cameron Musco, Christopher Musco, Rikhav Shah, John Urschel, and Nicholas West. Spectral density estimation for normal matrices. arXiv preprint arXiv:2605.31430 , 2026

    Pith/arXiv arXiv 2026

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    Topics in random matrix theory , volume 132

    Terence Tao. Topics in random matrix theory , volume 132. American Mathematical Society, 2023

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    High-dimensional probability, 2025

    Roman Vershynin. High-dimensional probability, 2025

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