Overfitted high-dimensional matrix factorizations via adaptive spectral shrinkage

David B. Dunson; Lorenzo Mauri

REVIEW 2 major objections 1 minor 12 references

Overfitted high-dimensional matrix factorizations via adaptive spectral shrinkage

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read EigenBayes performs spectral estimation of latent factors followed by adaptive empirical Bayes calibration to shrink superfluous components in overfitted high-dimensional factor models while delivering analytically tractable posteriors with

desk verdict EigenBayes pairs spectral factor estimates with adaptive EB to skip MCMC in overfitted models, but the adaptation and UQ claims under approximate low-SNR conditions need the full derivations to assess. read the letter →

arxiv 2606.19540 v1 pith:Y5HU2NAX submitted 2026-06-17 stat.ME stat.COstat.ML

Lorenzo Mauri , David B. Dunson This is my paper · ORCID

classification stat.MEstat.COstat.ML

keywords overfittedfactormodelshigh-dimensionalmatrixfactorizationadaptivespectralshrinkageempiricalBayesuncertaintyquantificationestimationlatentfactorscovariance

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 introduces EigenBayes as a fast alternative to Bayesian overfitted factor models for high-dimensional data. It estimates latent factors via spectral methods and calibrates hyperparameters through adaptive empirical Bayes, making the posterior factorize across outcomes and computable in closed form without MCMC. The approach is claimed to adapt automatically to the signal-to-noise ratio of each outcome and each latent dimension. It shrinks extra latent components to zero and is shown to have favorable asymptotic properties, with strong performance in simulations and a genomics example.

What carries the argument

The EigenBayes procedure, which combines spectral estimation of latent factors with adaptive empirical Bayes calibration of hyperparameters to produce a factorized, closed-form posterior.

What would settle it

A simulation experiment with misspecified or low-SNR factor models in which the EigenBayes posterior intervals fail to achieve nominal coverage or superfluous latent components remain unshrunk.

Watch

Extended reading notes

Core claim

We propose a much faster EigenBayes approach that provides valid uncertainty quantification, based on spectral estimation of latent factors and adaptive empirical Bayes calibration of key hyperparameters. The resulting posterior distribution factorizes across outcomes and is analytically tractable, bypassing Markov chain Monte Carlo. We show that EigenBayes adapts to the signal-to-noise ratio of each outcome and latent dimension, while shrinking superfluous latent components to zero. We establish favorable asymptotic properties.

Load-bearing premise

The adaptive empirical Bayes calibration of hyperparameters is assumed to produce valid uncertainty quantification and effective shrinkage even when the factor model holds only approximately and in low signal-to-noise regimes.

Editorial extensions

If this is right

The posterior factorizes across outcomes and is analytically tractable without MCMC.
Superfluous latent components are shrunk to zero while the method adapts to per-outcome and per-dimension signal-to-noise ratios.
Favorable asymptotic properties hold as dimensions grow.
The procedure outperforms state-of-the-art alternatives in numerical experiments and a genomics application.

Reading between the lines

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

The closed-form posterior could be used as a fast module inside larger pipelines that require repeated factor-model fits on very large matrices.
Analytic tractability may allow direct combination with other post-processing steps such as variable selection or prediction without additional sampling.
In applications like genomics the method suggests a route to routine uncertainty-aware covariance estimation on matrices too large for full Bayesian MCMC.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main thing to know is that Mauri and Dunson replace MCMC in overfitted factor models with a spectral estimation step followed by adaptive empirical Bayes hyperparameter calibration. This produces an analytically tractable posterior that factors across outcomes and is claimed to shrink extra latent dimensions while adapting to per-outcome signal strength.

The useful part is the speed improvement and the hybrid construction. Starting from spectral methods and then using data-driven EB to set the shrinkage and produce uncertainty quantification is a practical move for people who already like factor models but cannot run long chains. The abstract reports favorable asymptotics plus better performance than existing alternatives in simulations and a genomics application, which suggests the method delivers on the computational side.

The soft spot is the stress-test concern. When the low-rank-plus-diagonal structure holds only approximately and SNR is low, the initial spectral estimates can be inconsistent. It is not clear from the abstract whether the adaptive calibration step then still delivers calibrated credible sets or appropriate shrinkage, or whether it can over- or under-shrink. The paper asserts adaptation and valid UQ in exactly those regimes, so the strength of the result hinges on how the misspecification is handled in the proofs and whether the simulations include realistic low-signal cases.

This is for statisticians and applied researchers who need high-dimensional covariance tools and want something faster than full Bayesian sampling. A reader working on genomics or similar domains would find the empirical comparisons and the closed-form posterior worth seeing. It deserves peer review because the computational shortcut is concrete and the empirical results are presented as competitive, even if the theoretical robustness arguments will require close referee attention.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes EigenBayes, a fast alternative to Bayesian overfitted factor models for high-dimensional matrix factorizations. It uses spectral estimation of latent factors combined with adaptive empirical Bayes calibration of hyperparameters to produce an analytically tractable posterior that factorizes across outcomes, supplies valid uncertainty quantification without MCMC, adapts to per-outcome signal-to-noise ratio and latent dimension, and automatically shrinks superfluous components to zero. The paper claims favorable asymptotic properties together with strong performance in numerical experiments and a genomics application.

Significance. If the asymptotic guarantees and adaptation properties hold under the stated conditions, the work would supply a computationally scalable Bayesian procedure with built-in shrinkage and valid UQ for overfitted factor models, addressing a practical bottleneck in high-dimensional covariance estimation.

major comments (2)

[Abstract] Abstract: the assertion that adaptive empirical Bayes calibration yields 'valid uncertainty quantification' and 'adapts to the signal-to-noise ratio of each outcome' when the factor model holds only approximately is load-bearing for the central claim, yet the provided text supplies no explicit misspecification bounds or robustness analysis for the calibration step in low-SNR regimes.
[Abstract] Abstract (and method outline): the claim that the resulting posterior 'remains reliable' after spectral estimation followed by data-driven hyperparameter calibration requires a concrete statement of the conditions under which the empirical Bayes step does not induce circularity or miscalibration; without this, the adaptation and shrinkage properties cannot be assessed as non-circular.

minor comments (1)

[Abstract] Abstract: the description of empirical performance would be strengthened by explicit mention of the number of Monte Carlo replicates and any data-exclusion criteria used in the genomics application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract. We address each major comment below and will revise the manuscript to improve precision on the scope of the theoretical guarantees.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that adaptive empirical Bayes calibration yields 'valid uncertainty quantification' and 'adapts to the signal-to-noise ratio of each outcome' when the factor model holds only approximately is load-bearing for the central claim, yet the provided text supplies no explicit misspecification bounds or robustness analysis for the calibration step in low-SNR regimes.

Authors: The asymptotic results (Theorems 1--3) and adaptation properties are derived under the exact factor model. For approximate factor models the manuscript presents supporting numerical evidence in Section 5 across low-SNR and misspecified regimes, but does not supply explicit misspecification bounds on the empirical-Bayes calibration step. We will revise the abstract to qualify the claims and add a short discussion of this scope limitation in Section 3.4. revision: partial
Referee: [Abstract] Abstract (and method outline): the claim that the resulting posterior 'remains reliable' after spectral estimation followed by data-driven hyperparameter calibration requires a concrete statement of the conditions under which the empirical Bayes step does not induce circularity or miscalibration; without this, the adaptation and shrinkage properties cannot be assessed as non-circular.

Authors: The procedure is sequential: spectral estimates are obtained first and shown to be consistent (Proposition 1), after which the empirical-Bayes step calibrates hyperparameters using those estimates. Theorem 4 establishes consistency of the overall procedure under the paper's assumptions, so the calibration is not circular. We will add an explicit statement of these conditions to the abstract and method outline in the revision. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation relies on external spectral estimation and standard empirical Bayes calibration without self-referential reduction

full rationale

The abstract and provided excerpts describe EigenBayes as combining spectral estimation of factors with adaptive empirical Bayes hyperparameter calibration to yield an analytically tractable posterior. No equations or steps are quoted that reduce a claimed prediction or shrinkage property directly to the calibration inputs by construction. The method is presented as faster than MCMC-based overfitted factor models, with claimed adaptation and asymptotics that are not shown to be tautological. Self-citation is not load-bearing in the given text, and the approach is positioned against external benchmarks like existing Bayesian factor models. This is the common case of a self-contained proposal without exhibited circularity.

Assumptions & free parameters 1 free parameters · 0 assumptions · 0 invented entities

Abstract-only review; the method appears to rest on standard factor-model assumptions plus the unverified claim that empirical Bayes calibration yields valid posteriors. No explicit free parameters, axioms, or invented entities are named.

free parameters (1)

upper bound on latent dimension k
Specified as an input upper bound that the shrinkage is meant to prune.

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Cite this review

Pith. "Pith review of Overfitted high-dimensional matrix factorizations via adaptive spectral shrinkage." pith.science (2026). https://pith.science/paper/Y5HU2NAX

@misc{pith2026260619540,
  author       = {Pith},
  title        = {Pith review of: Overfitted high-dimensional matrix factorizations via adaptive spectral shrinkage},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5HU2NAX}},
  note         = {Machine review of arXiv:2606.19540}
}

abstract

Factor models are popular approaches for analyzing high-dimensional data to extract low-rank signals and estimate covariances. They decompose the covariance matrix as the sum of low-rank and diagonal components. A key issue is how to choose the latent dimension $k$, which is particularly challenging when the factor model only holds approximately and in low signal-to-noise scenarios. Bayesian overfitted factor models specify an upper bound on $k$ and rely on structured shrinkage priors to effectively remove extra components. Such approaches are popular and effective, but computationally expensive. We propose a much faster \texttt{EigenBayes} approach that provides valid uncertainty quantification, based on spectral estimation of latent factors and adaptive empirical Bayes calibration of key hyperparameters. The resulting posterior distribution factorizes across outcomes and is analytically tractable, bypassing Markov chain Monte Carlo. We show that \texttt{EigenBayes} adapts to the signal-to-noise ratio of each outcome and latent dimension, while shrinking superfluous latent components to zero. We establish favorable asymptotic properties and demonstrate strong empirical performance in numerical experiments and a genomics application, where EigenBayes outperforms state-of-the-art alternatives.

Figures

Figures reproduced from arXiv: 2606.19540 by the authors.

**Figure 2.** Relative Frobenius error in estimating the true covariance matrix under different sample [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Relative Frobenius error in estimating the true covariance matrix under different sample [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Relative Frobenius error in estimating the true covariance matrix under different sample [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Left panel: difference in out-of-sample log-likelihood between [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 3 canonical work pages

[1]

Ahn, S. C. & Horenstein, A. R. (2013), ‘Eigenvalue ratio test for the number of factors’,Econometrica 81(3), 1203–1227. Bai, J. (2003), ‘Inferential Theory for Factor Models of Large Dimensions’,Econometrica 71(1), 135–171. 14 Bai, J. & Ng, S. (2002), ‘Determining the number of factors in approximate factor models’,Econo- metrica70(1), 191–221. Bai, J. & ...

work page arXiv 2013
[2]

& Cho, H

Barigozzi, M. & Cho, H. (2018), ‘Consistent estimation of high-dimensional factor models when the factor number is over-estimated’,Electronic Journal of Statistics. Bhattacharya, A. & Dunson, D. B. (2011), ‘Sparse Bayesian infinite factor models’,Biometrika 98(2), 291–306. Blei, D., Kucukelbir, A. & McAuliffe, J. D. (2017), ‘Variational Inference: A Revie...

2018
[3]

& Stegle, O

Buettner, F., Pratanwanich, N., McCarthy, D., Marioni, J. & Stegle, O. (2017), ‘f-scLVM: scalable and versatile factor analysis for single-cell RNA-seq.’,Genome Biology18(212), 1–13. Chattopadhyay, S., Zhang, A. R. & Dunson, D. B. (2024), ‘Blessing of dimension in Bayesian inference on covariance matrices’,arXiv preprint arXiv:2404.03805. Chen, Y. & Li, X...

work page Pith review arXiv 2017
[4]

& Jakubzick, C

Desch, A., Murphy, K., Kedl, R., Lahoud, M., Caminschi, I., Shortman, K., Henson, P. & Jakubzick, C. (2011), ‘Cd103+ pulmonary dendritic cells preferentially acquire and present apoptotic cell- associated antigen’,The Journal of experimental medicine208, 1789–97. Doz, C., Giannone, D. & Reichlin, L. (2012), ‘A quasi–maximum likelihood approach for large, ...

2011
[5]

Overfitted high-dimensional matrix fac- torizations via adaptive spectral shrinkage

Mauri, L., Anceschi, N. & Dunson, D. B. (2025), ‘Spectral decomposition-assisted multi-study factor analysis’,arXiv preprint arXiv:2502.14600. Mauri, L. & Dunson, D. B. (2025a), ‘Factor pre-training in Bayesian multivariate logistic models’, Biometrikap. asaf056. Mauri, L. & Dunson, D. B. (2025b), ‘Inference on covariance structure in high-dimensional mul...

work page arXiv 2025
[6]

We proceed by showing consistency for the residual error variance

Putting everything together, with probability at least 1−𝑜(1), we have||𝐿 0 − ˆ𝐿||≲ √︁ 𝑛log𝑛, which implies ||𝐿 0 − ˆ𝐿|| ||𝐿 0|| ≲ √︁ log𝑛 √𝑛 , since||𝐿 0|| ≍𝑝≍𝑛. We proceed by showing consistency for the residual error variance. Let𝛿 2 𝑗 = 𝑣𝑠 2+| |𝑦 (𝑗) − ˆ𝑀 ˆ𝜆𝑗 | |2 𝑣+𝑛−2 . Using Lemma 12, we can write 𝛿2 𝑗 −𝜎 2 0 = 𝜎2 0 𝑛 𝑠2 𝑗 𝜎2 0 − (𝑛−𝑘) − 𝑘𝜎 2 0 𝑛 +...

2024
[7]

Then, with probability at least 1−𝑜(1), || ˜𝑍|| 2 ≲𝜌 2 𝑝||Ψ −1 𝑛, 𝑗 ||max 𝑗=1,..., 𝑝 𝜎2 𝑗 ≲ 𝑝 𝑛 ≍1, since||Ψ −1 𝑛,𝑙 || ≍ 1 𝑛, and, by Lemma 8, || ˆΛ||≲ √𝑝≍ √𝑛

NoteΠ(|| ˜Σ||< 𝐶) ≥1−𝑜(1), with probability at least 1−𝑜(1), by Lemma 11, whereΠ(·)denotes the probability measure induced by (6, 9). Then, with probability at least 1−𝑜(1), || ˜𝑍|| 2 ≲𝜌 2 𝑝||Ψ −1 𝑛, 𝑗 ||max 𝑗=1,..., 𝑝 𝜎2 𝑗 ≲ 𝑝 𝑛 ≍1, since||Ψ −1 𝑛,𝑙 || ≍ 1 𝑛, and, by Lemma 8, || ˆΛ||≲ √𝑝≍ √𝑛. Next, let𝐷 𝑗 =˜𝜎2 𝑗 −𝜎 2 0 and note|| ˜Σ−𝜎 2 0 𝐼𝑝 ||=max 𝑗=1,.....

2024
[8]

Hence, |𝑚 (𝑗)⊤ (𝑃−𝑃 0)𝑚 (𝑗 ′ ) | ≤ ∥𝑚 (𝑗) ∥ ∥𝑃0(𝑃−𝑃 0)𝑃0∥ ∥𝑚(𝑗 ′ ) ∥=O 𝑝𝑟 (1), and therefore𝑛 −1/2𝑚 (𝑗)⊤ (𝑃−𝑃 0)𝑚 (𝑗 ′ ) =𝑜 𝑝𝑟 (1). Next, let𝑃 (−𝑗 ′ ) =𝑈 (−𝑗 ′ ) 1:𝑘 𝑈 (−𝑗 ′ )⊤ 1:𝑘 , where 𝑈 (−𝑗 ′ ) 1:𝑘 ∈R 𝑛×𝑘 is the matrix of the left singular vectors associated to the leading-𝑘singular values of the matrix𝑌after deleting the𝑗 ′-th column. Then 𝑚 (𝑗)⊤ (𝑃...

2024

Show all 12 references

[9]

Then, with probability at least1−𝑜(1), we have √𝑛−𝐶 √ 𝑘≤𝑠 𝑙 (𝐹) ≤ √𝑛+𝐶 √ 𝑘,(𝑙=1,

A.2 Auxiliary lemmas Lemma 1.Let𝐹∈R 𝑛×𝑘 be a matrix with independent standard Gaussian entries. Then, with probability at least1−𝑜(1), we have √𝑛−𝐶 √ 𝑘≤𝑠 𝑙 (𝐹) ≤ √𝑛+𝐶 √ 𝑘,(𝑙=1, . . . , 𝑘), for a universal constant𝐶 <∞. Moreover, with probability at least1−𝑜(1), we have 𝐹⊤𝐹 𝑛 −...

2012
[10]

Let𝑀Λ ⊤ 0 =𝑈 0,1:𝑘 𝐷0,1:𝑘 𝑉 ⊤ 0,1:𝑘 be the singular value decomposition of the signal matrix, with𝑈 0 ∈R 𝑛×𝑘 having orthonormal columns

Lemma 6.Suppose Assumptions 1–4 hold. Let𝑀Λ ⊤ 0 =𝑈 0,1:𝑘 𝐷0,1:𝑘 𝑉 ⊤ 0,1:𝑘 be the singular value decomposition of the signal matrix, with𝑈 0 ∈R 𝑛×𝑘 having orthonormal columns. Let𝑈 1:𝑘 be the matrix of left singular vectors of𝑌associated to its leading𝑘singular values. Then, 𝑝𝑟...

1990
[11]

Lemma 8.Suppose Assumption 1–4 hold and𝐻=O (𝑛 2/5)

Hence, since𝐴has a positive eigengap between the𝑘-th and𝑘+1-th eigenvalues, with probability at least 1−𝑜(1), by Lemma 4, by the Davis–Kahan theorem applied to the leading𝑘-dimensional eigenspaces of𝐴and𝐴 (−𝑗 𝑗 ′ ), we have ∥𝑃−𝑃 (−𝑗 𝑗 ′ ) ∥ ≤𝐶 ∥𝐴−𝐴 (−𝑗 𝑗 ′ ) ∥ 𝑑2 𝑘 −𝑑 2 𝑘+1 ≲ ...

2024
[12]

29 Lemma 12.Let𝛿 2 𝑗 = 𝑣𝑠 2+| |𝑦 (𝑗) − ˆ𝑀 ˆ𝜆𝑗 | |2 𝑣+𝑛

Theorem 5 of Zhang & Zhou (2020) concludes the proof. 29 Lemma 12.Let𝛿 2 𝑗 = 𝑣𝑠 2+| |𝑦 (𝑗) − ˆ𝑀 ˆ𝜆𝑗 | |2 𝑣+𝑛 . Under Assumptions 1–4, we have 𝛿2 𝑗 = 𝑠2 𝑗 𝑛 +𝐹 𝑗, where 𝑠2 𝑗 𝜎2 0 ∼𝜒 2 𝑛−𝑘 andmax 𝑗=1,..., 𝑝 |𝐹𝑗 |≲ 1 𝑛2/5 , with probability at least1−𝑜(1). Proof of Lemma 12.As in...

2020

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Ahn, S. C. & Horenstein, A. R. (2013), ‘Eigenvalue ratio test for the number of factors’,Econometrica 81(3), 1203–1227. Bai, J. (2003), ‘Inferential Theory for Factor Models of Large Dimensions’,Econometrica 71(1), 135–171. 14 Bai, J. & Ng, S. (2002), ‘Determining the number of factors in approximate factor models’,Econo- metrica70(1), 191–221. Bai, J. & ...

work page arXiv 2013

[2] [2]

& Cho, H

Barigozzi, M. & Cho, H. (2018), ‘Consistent estimation of high-dimensional factor models when the factor number is over-estimated’,Electronic Journal of Statistics. Bhattacharya, A. & Dunson, D. B. (2011), ‘Sparse Bayesian infinite factor models’,Biometrika 98(2), 291–306. Blei, D., Kucukelbir, A. & McAuliffe, J. D. (2017), ‘Variational Inference: A Revie...

2018

[3] [3]

& Stegle, O

Buettner, F., Pratanwanich, N., McCarthy, D., Marioni, J. & Stegle, O. (2017), ‘f-scLVM: scalable and versatile factor analysis for single-cell RNA-seq.’,Genome Biology18(212), 1–13. Chattopadhyay, S., Zhang, A. R. & Dunson, D. B. (2024), ‘Blessing of dimension in Bayesian inference on covariance matrices’,arXiv preprint arXiv:2404.03805. Chen, Y. & Li, X...

work page Pith review arXiv 2017

[4] [4]

& Jakubzick, C

Desch, A., Murphy, K., Kedl, R., Lahoud, M., Caminschi, I., Shortman, K., Henson, P. & Jakubzick, C. (2011), ‘Cd103+ pulmonary dendritic cells preferentially acquire and present apoptotic cell- associated antigen’,The Journal of experimental medicine208, 1789–97. Doz, C., Giannone, D. & Reichlin, L. (2012), ‘A quasi–maximum likelihood approach for large, ...

2011

[5] [5]

Overfitted high-dimensional matrix fac- torizations via adaptive spectral shrinkage

Mauri, L., Anceschi, N. & Dunson, D. B. (2025), ‘Spectral decomposition-assisted multi-study factor analysis’,arXiv preprint arXiv:2502.14600. Mauri, L. & Dunson, D. B. (2025a), ‘Factor pre-training in Bayesian multivariate logistic models’, Biometrikap. asaf056. Mauri, L. & Dunson, D. B. (2025b), ‘Inference on covariance structure in high-dimensional mul...

work page arXiv 2025

[6] [6]

We proceed by showing consistency for the residual error variance

Putting everything together, with probability at least 1−𝑜(1), we have||𝐿 0 − ˆ𝐿||≲ √︁ 𝑛log𝑛, which implies ||𝐿 0 − ˆ𝐿|| ||𝐿 0|| ≲ √︁ log𝑛 √𝑛 , since||𝐿 0|| ≍𝑝≍𝑛. We proceed by showing consistency for the residual error variance. Let𝛿 2 𝑗 = 𝑣𝑠 2+| |𝑦 (𝑗) − ˆ𝑀 ˆ𝜆𝑗 | |2 𝑣+𝑛−2 . Using Lemma 12, we can write 𝛿2 𝑗 −𝜎 2 0 = 𝜎2 0 𝑛 𝑠2 𝑗 𝜎2 0 − (𝑛−𝑘) − 𝑘𝜎 2 0 𝑛 +...

2024

[7] [7]

Then, with probability at least 1−𝑜(1), || ˜𝑍|| 2 ≲𝜌 2 𝑝||Ψ −1 𝑛, 𝑗 ||max 𝑗=1,..., 𝑝 𝜎2 𝑗 ≲ 𝑝 𝑛 ≍1, since||Ψ −1 𝑛,𝑙 || ≍ 1 𝑛, and, by Lemma 8, || ˆΛ||≲ √𝑝≍ √𝑛

NoteΠ(|| ˜Σ||< 𝐶) ≥1−𝑜(1), with probability at least 1−𝑜(1), by Lemma 11, whereΠ(·)denotes the probability measure induced by (6, 9). Then, with probability at least 1−𝑜(1), || ˜𝑍|| 2 ≲𝜌 2 𝑝||Ψ −1 𝑛, 𝑗 ||max 𝑗=1,..., 𝑝 𝜎2 𝑗 ≲ 𝑝 𝑛 ≍1, since||Ψ −1 𝑛,𝑙 || ≍ 1 𝑛, and, by Lemma 8, || ˆΛ||≲ √𝑝≍ √𝑛. Next, let𝐷 𝑗 =˜𝜎2 𝑗 −𝜎 2 0 and note|| ˜Σ−𝜎 2 0 𝐼𝑝 ||=max 𝑗=1,.....

2024

[8] [8]

Hence, |𝑚 (𝑗)⊤ (𝑃−𝑃 0)𝑚 (𝑗 ′ ) | ≤ ∥𝑚 (𝑗) ∥ ∥𝑃0(𝑃−𝑃 0)𝑃0∥ ∥𝑚(𝑗 ′ ) ∥=O 𝑝𝑟 (1), and therefore𝑛 −1/2𝑚 (𝑗)⊤ (𝑃−𝑃 0)𝑚 (𝑗 ′ ) =𝑜 𝑝𝑟 (1). Next, let𝑃 (−𝑗 ′ ) =𝑈 (−𝑗 ′ ) 1:𝑘 𝑈 (−𝑗 ′ )⊤ 1:𝑘 , where 𝑈 (−𝑗 ′ ) 1:𝑘 ∈R 𝑛×𝑘 is the matrix of the left singular vectors associated to the leading-𝑘singular values of the matrix𝑌after deleting the𝑗 ′-th column. Then 𝑚 (𝑗)⊤ (𝑃...

2024