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REVIEW 3 major objections 4 minor 40 references

Eigenvector Overlaps of Random Covariance Matrices and their Submatrices

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For Gaussian covariance matrices and their row/column submatrices, this paper derives explicit asymptotic formulas for the rescaled mean squared singular-vector overlaps in the bulk, reducing in the zero-signal case to Cauchy-like…

desk verdict A genuinely new set of explicit overlap formulas for Wishart submatrices, built on a clean but not fully rigorous resolvent argument; the missing convergence proof is the price of admission. read the letter →

arxiv 2501.08768 v1 pith:HWSPCRVA submitted 2025-01-15 math.PR cond-mat.stat-mechq-fin.MF

Elie Attal , Romain Allez This is my paper · ORCID
classification math.PRcond-mat.stat-mechq-fin.MF MSC 60B2015B5260H10
keywords randommatrixtheorysingularvectorseigenvectoroverlapsWishartmatricesMarchenko-PasturdistributionprincipalcomponentanalysissubmatricesBrownianmotion
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 asks how much the singular vectors of a large Gaussian covariance matrix change when a macroscopic fraction of rows and columns is deleted: it derives the exact asymptotic value of the rescaled mean squared overlap between the two sets of singular vectors. The central result is a set of explicit formulas, valid for any deterministic initial matrix $A$, for right, left, and cross-overlaps of singular vectors in the bulk of the two spectra. In the null initial condition $A=0$, corresponding to the Marchenko-Pastur regime, the formulas reduce to Cauchy-like rational functions of the limiting eigenvalues $\lambda$ and $\mu$ and of the macroscopic ratios $q,\alpha,\beta$. These formulas quantify, eigenvector by eigenvector, how much principal-component structure survives when samples or features are removed, and thereby give a theoretical handle on PCA with missing data and on incremental PCA.

What carries the argument

The argument is carried by three double Stieltjes transforms of the squared overlaps, $S_V,S_U,S_W$, with poles at the eigenvalues of the full and truncated matrices; these resolvents smooth the microscopic oscillations and encode all overlap information. Using the Brownian dynamics of the singular values and vectors, the paper derives a deterministic coupled system of partial differential equations for the limiting resolvents (system (3.1)), and solves it explicitly by the method of characteristics, whose characteristics coincide with those of the Burgers equations satisfied by the spectral densities. An inversion formula then extracts the overlap functions $\bar V,\bar U,\bar W$ from the resolvents. In the zero-initial-matrix case the resolvents at time zero are simple rational functions, and the inversion reduces to elementary algebra involving the Hilbert transforms of the two Marchenko-Pastur densities.

What would settle it

For $A=0$, fix $q=0.9$, $\alpha=0.4$, $\beta=0.8$, $t=3$, choose two bulk quantiles $x,y$, and for $N=100,400,1600$ simulate many independent Gaussian matrices; compute the sample mean of $N\langle \tilde v_i|v_j\rangle^2$ and compare with the right side of (3.3). If the finite-$N$ values do not approach the predicted rational function, or approach a different denominator, the claimed limit is false.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the limiting rescaled mean squared overlaps are explicit, parameterized by the macroscopic ratios $q=\lim N/M$, $\alpha=\lim n/N$, $\beta=\lim m/M$ and the observation time $t$ (the variance of the added noise). For $A=0$, for $\lambda,\mu>0$ in the bulk, it derives $$N\mathbb{E}[\langle \tilde v_i|v_j\$rangle^{2}$]\to q\,\frac{(1-\$\alpha$)t\bar{\mu}+\$\alpha$(1-\$\beta$)t\bar{\$\lambda$}+(1-\$\alpha$\$\beta$)(\$\alpha$+1/q)$t^{2}$}{(1-\$\alpha$\$\beta$)^$2t^{2}$+q(\bar{\$\lambda$}-\bar{\mu})(\$\alpha$\$\beta$\bar{\$\lambda$}-\bar{\mu})},$$ with $\bar{\lambda}=\lambda-(1+1/q)t$ and $\bar{\mu}=\mu-(\alpha+\beta/q)t$; the left-overlap $\bar U$ and cross-overlap $\bar W$ satisfy analogous formulas with the same denominator, the cross-overlap having numerator $q(1-\alpha\beta)t\sqrt{\lambda\mu}$. The same machinery gives formulas for any deterministic initial matrix $A$, expressed through the initial values of three resolvents, so a nonzero $A$ changes the overlap formulas only through those initial resolvent data.

Load-bearing premise

The load-bearing premise is that the three resolvents converge to deterministic limits satisfying the closed PDE system (3.1); the paper shows several error terms vanish through variance bounds, but it gives no separate tightness or uniqueness proof for the limiting stochastic processes, and the inversion step would fail if that convergence did not hold.

Editorial extensions

If this is right

  • Deleting a fraction $1-\beta$ of the $M$ rows and $1-\alpha$ of the $N$ columns preserves singular-vector alignment at order $1/N$ in the bulk, with a universal prefactor fixed by $q,\alpha,\beta$ and the two limiting densities.
  • For fixed bulk eigenvalues $\lambda,\mu$, the overlap formulas depend on the deletion fractions only through the shifted variables $\bar{\lambda},\bar{\mu}$ and explicit pre-factors, so the structure of the decay is read off without simulation.
  • Incremental and missing-data PCA can compute the expected squared alignment between full-data and submatrix principal components for large Gaussian data directly from the rational expressions.
  • For nonzero initial matrix $A$, the method yields explicit overlaps whenever the initial resolvents $S_V,S_U,S_W(\cdot,\cdot,0)$ are computable, covering noisy observations of a deterministic signal.

Reading between the lines

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

  • Beyond the paper: the rational formulas could be turned into a finite-sample diagnostic for PCA with missing data, substituting observed bulk eigenvalues into the expression and comparing predicted with measured squared overlaps.
  • A testable extension is whether the leading Cauchy-like form survives for non-Gaussian entries with matching covariance; the Brownian-dynamics proof may change subleading terms while preserving the limiting rational function.
  • Because the null-space overlap formulas in (3.4) depend on the chosen basis for the null space, applications tracking zero singular values will need a basis-independent summary rather than the raw formulas.
  • The resolvent system identifies the limiting overlap field, not just its expectation, so one could push the same characteristic method to derive fluctuations or joint laws of the overlaps.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the overlaps between singular vectors of a rectangular Gaussian matrix X_t = A + N^{-1/2}B_t and of its m x n submatrix \tilde X_t, in the macroscopic regime N/M -> q, n/N -> alpha, m/M -> beta. It introduces the squared overlaps V, U, W and their double Stieltjes transforms S_V, S_U, S_W, derives Itô dynamics, and claims that in the scaling limit these transforms satisfy the deterministic PDE system (3.1). Solving this system and applying an inversion formula yields explicit formulas for the limiting rescaled mean squared overlaps for a general initial matrix A, formulas (3.2), which in the case A=0 reduce to the Marchenko-Pastur expressions (3.3). The paper also reports numerical simulations that agree well with the A=0 formulas.

Significance. The main formulas are explicit and parameter-free: q, alpha, beta and t are model inputs, not fitted constants, and the A=0 result gives Cauchy-like closed forms for bulk overlaps that are directly testable by simulation. This extends the authors' earlier Wigner/submatrix analysis to Wishart-type covariance matrices and to left/right/cross overlaps, and addresses a natural question with applications to PCA of submatrices, missing data and incremental PCA. The algebraic resolution of the PDE system and the appendix computations are detailed and internally coherent. The main weakness is that the rigorous justification of the convergence step is missing, so the central claim currently rests on an ansatz.

major comments (3)
  1. [Section 3.1 and Appendix D] The load-bearing step is the claim that the empirical double resolvents S_V^(N), S_U^(N), S_W^(N) converge to deterministic limits and that these limits solve (3.1). Section 3.1 only states that 'We typically expect these quantities to converge to deterministic integrals', and Appendix D derives the PDE by replacing finite-N products such as S_V^(N)S_W^(N) by products of limiting values, for example in the convergence of I_mu_lambda to 2 S_W S_V. No tightness, uniform-integrability, or uniqueness argument is supplied for the processes, and no variance bound for S^(N) - E S^(N) is given. Since the inversion formulas for \bar V, \bar U, \bar W are valid only if the empirical resolvents actually converge to the objects being inverted, this gap is load-bearing for (3.2) and (3.3).
  2. [Appendix D, paragraph on Brownian terms] The Borel-Cantelli argument for the martingale increments dI_w is not sufficient. The bound E|dI_w|^2 = O(1/N^2) controls the square of the increment over an infinitesimal time interval at a fixed time, not the time-uniform convergence of the integrated process needed to identify the limiting PDE. To drop these terms in the limit equation one would need a maximal inequality or a convergence theorem for the semimartingales S^(N).
  3. [Section 2 and Appendix B] For the general-initial-condition formulas (3.2), the derivation also assumes that the empirical eigenvalue densities converge and satisfy the limiting Burgers equations (2.4) and (2.6). Appendix B computes the Itô formula and then passes to the limit formally. For A=0 this is known Marchenko-Pastur theory, but for arbitrary A the propagation of convergence of the empirical spectral measure is not proved, so the 'any initial matrix A' claim is not fully established.
minor comments (4)
  1. [Section 3.2] The text states that the three expressions in (3.3) 'are still valid in any other case', but the derivation is carried out only under M >= N and m >= n; please state the precise symmetry argument or mark this as a conjecture.
  2. [Section 3.1, after (3.1)] The notation S0(t) introduced after (3.1) is ambiguous because S0 is used for each of the three transforms; use S0_V, S0_U, S0_W in the displayed solution.
  3. [Figure 1] The figure would be more informative with error bars or a quantitative discrepancy measure; the caption's 'fit is excellent' is not supported by numbers.
  4. [Equations (3.3)] The indexing in the limit statements (3.3) should make the bulk positions explicit: i_N/n -> x and j_N/N -> y, with mu and lambda the limiting quantile values, rather than merely mu_iN -> mu.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: formulas (3.3) follow from explicit t=0 resolvents and standard Marchenko-Pastur data; the unproven resolvent convergence is a correctness gap, not a circular reduction.

full rationale

The derivation of (3.3) is self-contained and free of fitted inputs. In the Marchenko-Pastur case, the limiting resolvents are obtained by solving the coupled system (3.1) with explicit initial values SV(z,~z,0)=alpha/(z~z), SU(z,~z,0)=beta/(q z~z), SW(z,~z,0)=0, which follow from X0=0 and the chosen orthonormal bases, and with spectral densities and Stieltjes transforms given by the standard Marchenko-Pastur law. The parameters q, alpha, beta and t are model inputs, not fitted constants, so the headline overlap formulas are genuine predictions rather than re-parameterized data. The inversion formulas are attributed to the independent work [10] (Bun-Bouchaud-Potters) and to the authors' previous paper [4] only as a methodological template; the resolvent construction is re-derived here in Appendices C-D, so the self-citation is not load-bearing. The paper's main limitation is that Section 3.1 states 'We typically expect these quantities to converge to deterministic integrals,' and Appendix D proves bounds on selected Brownian and remainder terms but does not supply a full tightness/uniqueness argument identifying finite-N resolvents with the solution of (3.1). This is a missing convergence proof and therefore a correctness risk, but it is not circularity: no final formula is equal by construction to an input, and no fitted parameter is renamed as a prediction. Independent numerical simulations reported in Figure 1 provide external support for the closed forms.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central formulas have no fitted constants; q, alpha, beta and t are model inputs. The derivation relies on external results from Bru, on the inversion formula from [10], and on a partially proved self-averaging assumption for the three resolvents. No new physical entities are introduced.

assumptions (4)
  • domain assumption The empirical spectra of A^T A and of the submatrix initial condition converge to continuous densities rho(lambda,0) and tilde_rho(mu,0).
    Section 2, just before the Stieltjes transforms are introduced. This assumption defines the bulk regime and is needed for the quantile functions and inversion formulas.
  • standard math The eigenvalue and eigenvector dynamics of X_t are given by Bru's equations (2.1)-(2.2), including the zero-eigenvalue convention for the left singular vectors.
    Bru's 1989 theorem is used as a black box, and the same form is asserted for the truncated matrix. This is a cited external result, not reproved here.
  • ad hoc to paper The double resolvent processes S_V, S_U, S_W self-average to deterministic limits, and products of empirical limits converge to products of the limits.
    Section 3.1 expects convergence, and Appendix D derives the PDE conditional on that convergence. Variance bounds are given for some error terms, but a full tightness or uniqueness argument is not supplied.
  • standard math The double-Stieltjes inversion formula from [10] recovers bar_V, bar_U, bar_W from the limiting resolvents.
    The inversion formula is cited from Bun, Bouchaud and Potters and was also used in [4]. It is not proved in this paper.

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Pith. "Pith review of Eigenvector Overlaps of Random Covariance Matrices and their Submatrices." pith.science (2026). https://pith.science/paper/HWSPCRVA

@misc{pith2026250108768,
  author       = {Pith},
  title        = {Pith review of: Eigenvector Overlaps of Random Covariance Matrices and their Submatrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWSPCRVA}},
  note         = {Machine review of arXiv:2501.08768}
}
abstract

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as well as $n \,/\, N$ converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix $A\,$. When it is null, this corresponds to the Marchenko-Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.

Figures

Figures reproduced from arXiv: 2501.08768 by the authors.

Figure 3
Figure 3. shows a comparison of these formulas with simulated rescaled mean squared [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 1
Figure 1. Comparison of our formulas for V , ¯ U¯ and W¯ with numerical simulations of N E [Vij (t)] (red plain curve for theory and red circles for data), N E [Uij (t)] (blue plain curve for theory and blue triangles for data) and N E [Wij (t)] (green plain curve for theory and green squares for data) for M = 300 , q = 0.9 , α = 0.4 , β = 0.8 and t = 3 as a function of λ for a fixed µ = µ(x, t). Left: x = 0.9 . Middle: x = 0… view at source ↗

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