High-dimensional ridgeless least squares interpolation under spiked covariance structures

Dandan Jiang; Zhijun Liu

REVIEW 5 major objections 6 minor 20 references

High-dimensional ridgeless least squares interpolation under spiked covariance structures

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Under a generalized spiked covariance model, the out-of-sample prediction risk of the ridgeless least-squares estimator has a sharp asymptotic limit determined by spike eigenvalues, aspect ratio, and the alignment of the regression vector…

desk verdict Plausible multi-spike ridgeless risk formula, but the central theorem is not actually proven and the overfitting classification runs outside the assumptions. read the letter →

arxiv 2608.07281 v1 pith:GEODB6O5 submitted 2026-08-07 math.ST cs.LGstat.MLstat.TH

Zhijun Liu , Dandan Jiang This is my paper · ORCID

classification math.STcs.LGstat.MLstat.TH MSC 60B2060F05

keywords predictionrisklinearspectraldistributionrandommatrixtheoryridgelessleast-squaresestimatorspikedcovariancemodelbenignoverfittingdoubledescenttarget-spikealignment

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

This paper aims to prove that, in proportional-limit high-dimensional regression (p/n to gamma), the prediction risk of the minimum-norm least-squares interpolator is fully characterized by the spiked covariance spectrum, the aspect ratio, and the squared alignment between the regression vector $\beta$ and each spike eigenvector. In the overparameterized regime gamma > 1, the risk limit is an explicit formula; in the underparameterized regime gamma < 1, the limit is $sigma^{2}$ gamma/(1-gamma) and is insensitive to spikes. The paper further derives a taxonomy of benign, tempered, and catastrophic overfitting driven by the growth of aggregate spike strength relative to gamma, and shows that alignment with spike directions is always detrimental. These results matter because they explain when latent factor structure in features helps or hurts generalization of interpolating models.

What carries the argument

The machinery is the generalized spiked covariance model Sigma = U diag(alpha_1,...,alpha_M,1,...,1) U^T together with random-matrix-theory limits for the sample spike eigenvalues and eigenvectors. The paper imports three limit results: the almost-sure limit of the sample spiked eigenvalues $\varphi$(alpha_k) = alpha_k (1 + gamma/(alpha_k - 1)), the almost-sure limit of the squared overlap between the sample and population spike eigenvectors, and a lemma (Lemma 4.4) giving the spike eigenvalue and inverse-resolvent limits when M grows as o($n^{{1/4}}$) under finite fourth moments. These limits are used to decompose the risk functional tr(Sigma_hat^+ Sigma) and $\beta$^T Pi Sigma Pi $\beta$ into bulk and spike components, which yields the explicit risk formula.

What would settle it

Run the M=1 case with gamma=2, alpha_1=4, $r^{2}$=$sigma^{2}$=1, and $\beta$ = u_1 so that <u_1,$\beta$>^2=1; Theorem 3.1 predicts the risk converges to (3)(1/2)^2 + (1/2) + (1)(1/4 + 1) = 0.75 + 0.5 + 1 = 2.25 as n,p -> infinity. If finite-n Monte Carlo estimates of the min-norm estimator's out-of-sample risk do not approach 2.25 (e.g., they saturate elsewhere or fail to converge), the formula is refuted.

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Extended reading notes

Core claim

Theorem 3.1 asserts that as n,p -> infinity with p/n -> gamma, the conditional prediction risk of the ridgeless estimator satisfies R_X(beta_hat; $\beta$) - [sum_{i=1}^M (alpha_i - 1)(1 - 1/gamma)^2 <u_i,$\beta$>^2 + (1 - 1/gamma) $r^{2}$ + $sigma^{2}$/(gamma-1) ( (sum_{j=1}^M alpha_j)/p + (p-M)/p )] -> 0 in probability, for gamma > 1, and R_X -> $sigma^{2}$ gamma/(1-gamma) for gamma < 1. The display combines the bias term, which carries the alignment dependence through <u_i,$\beta$>^2, and the variance term, which depends on the average spike strength. The paper interprets this formula as showing that spike structure modifies the variance only when the aggregate spike strength is of order p or larger, and modifies the bias whenever $\beta$ is not orthogonal to the spike eigenvectors.

Load-bearing premise

The main theorem's diverging-M case rests on an imported lemma (Lemma 4.4) that is not proved in this paper; if that lemma's spike eigenvalue and inverse-resolvent limits fail under the stated finite-fourth-moment condition, the central risk formula's growing-M version does not follow.

Editorial extensions

If this is right

For gamma > 1, the prediction risk limit is an explicit function of the spike eigenvalues, the aspect ratio, and the squared alignment <u_i,beta>^2; the formula is sharp enough to be used as a benchmark for finite-sample behavior.
When the aggregate spike strength is o(p), the variance contribution matches the isotropic covariance case sigma^2/(gamma-1), and only the bias term is changed by the spikes.
Any nonzero alignment between beta and a spike direction increases the asymptotic risk, so restricting interpolation to the orthogonal complement of the spike eigenspace is the paper's suggested way to control risk.
The overfitting taxonomy says benign overfitting occurs when beta is orthogonal to the spike directions and the aggregate spike strength grows slower than gamma; if beta has nonzero alignment, tempered or catastrophic overfitting follow under spike growth conditions.
The underparameterized regime gamma < 1 is spike-insensitive: the risk tends to sigma^2 gamma/(1-gamma), independent of both the spectrum and the alignment.

Reading between the lines

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

A practical extension the authors do not spell out: if one could estimate the spike subspace from unlabeled features, projecting the design (or orthogonalizing beta) before fitting should lower interpolation risk; this is testable with factor-model datasets.
The imported lemma constrains M to o(n^{1/4}) under finite fourth moments; a self-contained proof or a numerical stress test at M near n^{1/4} would indicate how much of the diverging-M claim is carried by external results.
Theorem 4.1's gamma -> infinity classification formally needs alpha_i > 1 + sqrt(gamma), which cannot hold for bounded or slowly growing spikes; reading it literally, the regime labels apply only to spikes growing at least like sqrt(gamma), an implicit condition that should be stated.
The formula's structure suggests a more general principle: the risk impact of any covariance direction is proportional to (alpha - 1) times the signal squared along that direction; this could be extended to full non-spiked spectra by integration, a natural next step the authors do not pursue.

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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 goes after the right problem and the formulas look right. The multi-spike generalization of ridgeless least-squares risk, with M growing as o(n^{1/4}) and only finite fourth moments, is a genuine extension of the single-spike results of Li and Sonthalia. The formula reduces correctly to the isotropic case, the single-spike case, and the known gamma<1 limit, and the simulations line up with the displayed approximations. That is real value.

The problem is that Theorem 3.1, the central claim, is stated without proof. The appendix only imports lemmas from other papers. The variance limit in (3.9) is not a direct consequence of the cited Lemma 4.4; you need a bulk-eigenvector delocalization step, essentially replacing v_k^T Sigma v_k by tr(Sigma)/p for bulk eigenvectors, and no lemma or argument supplies that. The theorem claims almost sure convergence while Lemma 4.4 is only convergence in probability. So the paper currently does not establish its main result.

Theorem 4.1 is also stated in a regime the assumptions exclude. Assumption 2 requires alpha_i > 1 + sqrt(gamma). Taking gamma to infinity while allowing bounded or slowly growing spikes means those spikes eventually violate the assumption. The benign/tempered/catastrophic classification may be right, but as a theorem it is not justified.

There are also minor slips: the keywords include "prediction disk," the Figure 5 caption omits the sum variable, and the text around Corollary 3.2 has some imprecise statements about divergent spikes and o(p). These are small; the two gaps above are what matter.

I think the core formula is believable and the paper would be a useful reference once the proof is actually written. But in its current form, the main theorem and the overfitting classification are unsupported. A serious referee could assess whether the missing delocalization step can be supplied, so I would not desk-reject the topic; but the paper is not close to publishable as is.

Recommendation: send it to peer review only if you expect the authors to fill in the proof; otherwise it will be a waste of referee time. For a reading group, it is a decent case study in what counts as a proof gap, but I would not cite it yet.

Referee Report

5 major / 6 minor

Summary. The paper studies the out-of-sample prediction risk of the min-norm (ridgeless) least squares estimator under a spiked covariance model in the proportional asymptotics p/n -> γ. The main result, Theorem 3.1, claims that for γ < 1 the risk converges to σ²γ/(1-γ), and for γ > 1 gives explicit bias and variance limits that depend on the spike eigenvalues, the aspect ratio, and the squared alignments ⟨u_i, β⟩² of the regression vector with the spike eigenvectors. Section 4 uses these limits to classify benign, tempered, and catastrophic overfitting as γ → ∞ and includes simulation figures. The central theorem is stated without proof; the appendix only reproduces imported lemmas from other papers.

Significance. If the claimed formula were established, the paper would be a useful extension of the isotropic analysis of Hastie et al. (2022) to multi-spike covariance structures under finite fourth moments, and the alignment-driven phase classification would be a genuinely new contribution. The paper also verifies consistency with known isotropic and single-spike special cases and provides illustrative simulations. However, the central result is not proved, and the variance formula requires a bulk-eigenvector delocalization statement that is neither stated nor supplied. The significance is therefore conditional on a proof that is not present in the current manuscript.

major comments (5)

[Section 3, Theorem 3.1] Theorem 3.1, the central claim of the paper, is stated without proof. No derivation of the bias limit (3.8) or the variance limit (3.9) is given in Section 3, and the appendix contains only imported Lemmas 4.2–4.4, which do not by themselves establish the risk limits. The manuscript accordingly does not currently establish its main result.
[Eq. (3.9)] The variance formula is not a direct consequence of Lemma 4.4 as stated. Lemma 4.4 controls sample spiked eigenvalues and inverse-resolvent quantities, but passing from tr(Σ̂⁺Σ)/n to σ²/(γ-1) (Σ_{j=1}^M α_j/p + (p-M)/p) requires replacing v_kᵀ Σ v_k by tr(Σ)/p for the bulk sample eigenvectors, i.e., a bulk-eigenvector delocalization result relative to the spike directions. No such lemma appears in the paper, so (3.9) lacks a load-bearing derivation.
[Theorem 3.1 and Lemma 4.4] The convergence modes are inconsistent. Theorem 3.1 claims almost sure convergence, while Lemma 4.4 is quoted as convergence in probability, and no argument is provided to upgrade the mode. Additionally, (3.8) and (3.10) contain O_p(1/p) and O_p(⟨u_i,β⟩/√p) remainders inside an almost sure limit statement; these terms need to be shown to be o(1) almost surely (with uniform rates in i), otherwise the displayed convergence is not well-formed.
[Theorem 4.1 and Assumptions 1–2] Theorem 4.1 considers γ → ∞ while Assumption 1 fixes p/n → γ > 0 and Assumption 2 requires α_i > 1 + √γ. For any fixed or bounded α_j, the condition α_j > 1 + √γ fails eventually as γ → ∞, so classification cases with constant-order α_j are not covered by the stated assumptions. The theorem needs a separate asymptotic regime with compatible assumptions on the growth of the spikes relative to γ.
[Section 4.1, Proposition 4.1] The overfitting taxonomy takes a second limit γ → ∞ after defining R_γ as an n,p limit at fixed γ, but the paper does not state a joint limit or a uniformity condition that would justify exchanging these limits. Since the O_p remainders in (3.8) may depend on γ, the double-limit argument is not justified without additional uniformity or rate control.

minor comments (6)

[Section 1] Equations (1.3) and (1.5) are identical and should not both be displayed; one should be removed.
[Keywords] The keyword phrase 'prediction disk' appears to be a typo; it should probably be 'prediction risk'.
[Figures 5] The caption of Figure 5(b) says P_{i=1}^5 α_i = 200 while the text in Section 4.2 says 500; also the caption of Figure 5(a) appears to omit the summed quantity in one place.
[Section 4.1] The phrasing 'not orthogonal to all vectors in set {u_j}' and 'not orthogonal to any vector in set {u_j}' is ambiguous; the intended quantifiers should be spelled out.
[Section 3, Eqs. (3.8) and (3.10)] The O_p symbols inside the displayed limits should be replaced by explicit remainder terms with stated rates and uniformity in i, since the current notation is non-standard in an almost sure convergence statement.
[Theorem 4.1] The phrase 'the rate at which γ → ∞' is not formal. If γ_n = p/n is intended to diverge, the paper should define the joint asymptotic regime and state the corresponding versions of Assumptions 1–3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the risk limit is assembled from independent external random-matrix lemmas; the only self-citation is an uncited reference-list entry and is not load-bearing. Theorem 3.1 lacks a proof, but that is a completeness gap, not circularity.

full rationale

The paper's central claim, Theorem 3.1 (equations (3.7)-(3.10)), states asymptotic limits for the bias and variance of the ridgeless least-squares estimator under a generalized spiked covariance model. No circular step can be exhibited: the alignment terms <u_i, beta>^2, the spike strengths alpha_i, the aspect ratio gamma, the noise level sigma^2, and the signal norm r^2 are all treated as exogenous inputs and are not fitted to the paper's simulations or renamed predictions. The stated limits are assembled from independent external results — Silverstein's LSD equation, Johnstone and Yang (2018) eigenvector-overlap limits, Bai and Yao (2012) spike eigenvalue limits, and Hu et al. (2026) diverging-spike eigenvalue limits — none of which are equivalent to the target risk by construction. The reduction to Hastie et al. (2022) when alpha_i = 1 is an external consistency check, not circularity. Theorem 4.1 is obtained by substituting gamma -> infinity into (3.10), so the overfitting classification is a consequence of the risk formula rather than an input to it. The only self-citation, Jiang and Bai (2021), appears in the reference list but is not cited in the body and is not load-bearing. The manuscript has genuine non-circularity concerns: Theorem 3.1 is asserted without a proof in Section 3, the Appendix only lists borrowed lemmas (Lemma 4.2-4.4), and the gamma -> infinity classification in Theorem 4.1 conflicts with Assumption 2's condition alpha_i > 1 + sqrt(gamma). These are correctness/completeness issues, not circularity. Since no derivation step reduces to its own inputs, the circularity score is 0.

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

No free parameters are fitted: the spike eigenvalues, aspect ratio, and alignment are inputs treated as given. All axioms are either standard random matrix theory or domain assumptions on the data-generating process; no new entities are postulated.

assumptions (6)

domain assumption Proportional asymptotics p/n -> gamma in (0,infty).
Assumption 1; defines the asymptotic regime and excludes vanishing or exploding aspect ratios.
domain assumption Population covariance is a generalized spiked model with alpha_i > 1 + sqrt(gamma), M fixed or M = o(n^{1/4}), finite fourth moments.
Assumption 2; the separation condition alpha_i > 1 + sqrt(gamma) is required for the eigenvector alignment and eigenvalue limits in Lemmas 4.2-4.4.
domain assumption The spectral distribution H_n of Sigma converges to a limit H.
Assumption 3; needed to invoke Silverstein's equation (2.6).
standard math Silverstein's LSD equation characterizes the bulk spectrum of the sample covariance.
Cited Silverstein (1995), equation (2.6); standard random matrix theory used for the variance term.
domain assumption Sample eigenvector overlaps with population spike directions converge to the Johnstone-Yang limit.
Lemma 4.2 (Johnstone and Yang 2018); used to compute bias contributions along spike directions.
domain assumption The diverging-M eigenvalue and inverse-resolvent limits of Hu et al. (2026) hold under finite fourth moments with M = o(n^{1/4}).
Lemma 4.4; load-bearing for the M -> infinity case of Theorem 3.1, but no proof is given in this paper.

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Cite this review

Pith. "Pith review of High-dimensional ridgeless least squares interpolation under spiked covariance structures." pith.science (2026). https://pith.science/paper/GEODB6O5

@misc{pith2026260807281,
  author       = {Pith},
  title        = {Pith review of: High-dimensional ridgeless least squares interpolation under spiked covariance structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GEODB6O5}},
  note         = {Machine review of arXiv:2608.07281}
}

abstract

This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension $p$ and the sample size $n$ grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the number of spiked eigenvalues may remain finite or increase with $n$, and the spiked eigenvalues may be bounded or diverge at arbitrary rates. Beyond characterizing the impact of covariance spectra, we reveal a new mechanism underlying benign overfitting: the prediction behavior of ridgeless interpolation is fundamentally governed by the alignment between the regression coefficient $\boldsymbol\beta$ and the spiked eigenspaces of the population covariance matrix. In particular, we show that the signal energy distributed along latent spike directions determines whether interpolation leads to benign, tempered, or catastrophic overfitting. Our theoretical framework establishes sharp prediction risk limits under minimal moment conditions, requiring only finite fourth moments rather than Gaussianity. We characterize how the number, strength, and geometric structure of the spikes jointly influence the double-descent phenomenon. These results provide a unified understanding of when latent covariance structures facilitate or hinder generalization in overparameterized regression.

Figures

Figures reproduced from arXiv: 2608.07281 by the authors.

**Figure 2.** Left: The asymptotic risk surface varying with [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Left: Asymptotic risk surfaces for the bias (blue) and variance (orange) terms as func [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Left: The asymptotic risk surface varying with [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: The asymptotic risk curves as a function of the aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Determining the number of factors in approximate factor models

Jushan Bai and Serena Ng. Determining the number of factors in approximate factor models. Econometrica, 70 0 (1): 0 191--221, 2002

2002

[2] [2]

Large sample covariance matrices without independence structures in columns

Zhidong Bai and Wang Zhou. Large sample covariance matrices without independence structures in columns . Statistica Sinica,18 0: 0 425 -- 442, 2008

work page 2008

[3] [3]

On sample eigenvalues in a generalized spiked population model

Zhidong Bai and Jianfeng Yao. On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis, 106: 0 167--177, 2012

work page 2012

[4] [4]

Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis

Zhidong Bai, Kwok Pui Choi and Yasunori Fujikoshi. Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis. The Annals of Statistics, 46: 0(3) 1050--1076, 2018

work page 2018

[5] [5]

Silverstein

Jinho Baik and Jack W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 97 0 (6): 0 1382--1408, 2006

2006

[6] [6]

Benign overfitting in linear regression

Peter L Bartlett, Philip M Long, Gábor Lugosi and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30063–30070, 2020

work page 2020

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Reconciling modern machine-learning practice and the classical bias–variance trade-off

Mikhail Belkin, Daniel Hsu, Siyuan Ma and Soumik Mandal. Reconciling modern machine-learning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116: 0(32) 15849--15854, 2019

work page 2019

[8] [8]

Two models of double descent for weak features

Mikhail Belkin, Daniel Hsu and Ji Xu. Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2: 0(4) 1167--1180, 2020

work page 2020