Inference on covariance structure in high-dimensional multi-view data

David B. Dunson; Lorenzo Mauri

arxiv: 2509.02772 · v2 · submitted 2025-09-02 · 📊 stat.ME · stat.CO· stat.ML

Inference on covariance structure in high-dimensional multi-view data

Lorenzo Mauri , David B. Dunson This is my paper

Pith reviewed 2026-05-18 19:03 UTC · model grok-4.3

classification 📊 stat.ME stat.COstat.ML

keywords covariance estimationmulti-view datahigh-dimensional statisticsconjugate priorsposterior contractionfactor modelsmulti-omics integration

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The pith

Spectral decompositions align latent factors in multi-view data so that conjugate priors produce closed-form posteriors for high-dimensional covariance estimation.

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

The paper develops a Bayesian approach to covariance estimation when data arrive in multiple views, each potentially carrying both shared and view-specific structure. Spectral methods first recover and align the latent factors that appear in any view. Given those factors, jointly conjugate priors are placed on loadings and variances so that the full posterior factors exactly into independent normal-inverse-gamma distributions, one per variable. This factorization removes the need for MCMC or variational approximations while delivering posterior contraction rates and central-limit results for the resulting estimators as dimension grows.

Core claim

Conditionally on latent factors recovered by spectral decomposition, jointly conjugate priors for factor loadings and residual variances yield a posterior that is exactly a product of normal-inverse-gamma distributions across variables; the resulting procedure therefore admits direct posterior sampling and satisfies increasing-dimension posterior contraction together with central limit theorems for point estimators.

What carries the argument

Spectral decomposition that estimates and aligns latent factors active in at least one view, which then conditions the choice of jointly conjugate priors for loadings and variances to obtain the product posterior.

If this is right

Posterior computation reduces to direct sampling from independent normal-inverse-gamma distributions for each variable.
Uncertainty quantification remains accurate rather than underestimated by variational approximations.
Point estimators obey central limit theorems, permitting standard-error-based inference in high dimensions.
The procedure scales to integrating four or more high-dimensional views, as demonstrated on multi-omics cancer data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same spectral-plus-conjugate construction could be tested on single-view factor models if the alignment step is replaced by a suitable initialization.
Robustness checks could examine performance when some latent factors are weak or when views have mismatched sample sizes.
The closed-form posterior might be combined with variable selection penalties to produce sparse multi-view covariance estimates.

Load-bearing premise

Spectral decompositions can reliably recover and align the latent factors that are active in at least one view from the observed multi-view data.

What would settle it

A simulation or real dataset in which the spectral estimates fail to align the active factors and the resulting posterior intervals or point estimators do not contract at the claimed rate.

Figures

Figures reproduced from arXiv: 2509.02772 by David B. Dunson, Lorenzo Mauri.

**Figure 2.** Figure 2: Comparison of different methods in terms of normalized root mean squared error of the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Within-view correlation plots inferred by [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Across-view correlation plots inferred by [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

read the original abstract

This article focuses on covariance estimation for multi-view data. Popular approaches rely on factor-analytic decompositions that have shared and view-specific latent factors. Posterior computation is conducted via expensive and brittle Markov chain Monte Carlo (MCMC) sampling or variational approximations that underestimate uncertainty and lack theoretical guarantees. Our proposed methodology employs spectral decompositions to estimate and align latent factors that are active in at least one view. Conditionally on these factors, we choose jointly conjugate prior distributions for factor loadings and residual variances. The resulting posterior is a simple product of normal-inverse gamma distributions for each variable, bypassing MCMC and facilitating posterior computation. We prove favorable increasing-dimension asymptotic properties, including posterior contraction and central limit theorems for point estimators. We show excellent performance in simulations, including accurate uncertainty quantification, and apply the methodology to integrate four high-dimensional views from a multi-omics dataset of cancer cell samples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Closed-form posteriors via spectral factors and conjugates for multi-view covariance, but the asymptotics need explicit error bounds on the plug-in step.

read the letter

The main point is that this paper develops a Bayesian procedure for covariance estimation in high-dimensional multi-view data that achieves a closed-form posterior by conditioning on spectral estimates of the latent factors and using conjugate priors. This setup lets them skip MCMC sampling entirely and get a product of normal-inverse-gamma distributions per variable. They align factors active in at least one view through spectral decomposition, then apply jointly conjugate priors for loadings and residual variances. The authors also state asymptotic results on posterior contraction and central limit theorems as dimensions grow, report strong simulation performance including uncertainty quantification, and show an application to a four-view multi-omics cancer dataset. The novelty is in the specific pairing that delivers the exact factorization while still attempting theoretical support, which differs from the usual MCMC or variational routes in multi-view factor models. It performs well on the computational side by making posterior work direct and by including a relevant real-data example. A potential issue is whether the asymptotic guarantees survive the plug-in of spectral factor estimates. The theory appears derived conditional on those estimates, so without uniform bounds showing the spectral error is negligible relative to the contraction rates when both dimension and number of views increase, the claims could be more approximate than exact. The abstract states the results, but the strength depends on the details of those bounds. This work targets statisticians and applied researchers working on data integration in genomics or similar areas. Readers who need scalable Bayesian covariance tools with uncertainty quantification and some theory would find it useful. It has enough structure and practical claims to deserve a serious referee rather than a desk reject. I would send it out for peer review, asking referees to focus on the asymptotic error control and the simulation details.

Referee Report

2 major / 2 minor

Summary. The paper develops a covariance estimation procedure for high-dimensional multi-view data. Latent factors shared or active across views are estimated and aligned via spectral decomposition. Conditionally on these point estimates, jointly conjugate normal-inverse-gamma priors are placed on loadings and residual variances, yielding an exact product-of-NIG posterior that avoids MCMC. The authors establish increasing-dimension posterior contraction and central limit theorems for the resulting point estimators, demonstrate accurate uncertainty quantification in simulations, and apply the method to a four-view multi-omics cancer dataset.

Significance. If the central claims hold, the work supplies a computationally tractable, theoretically supported alternative to MCMC or variational methods for multi-view covariance inference. The closed-form posterior factorization and the increasing-dimension contraction/CLT results are substantive strengths; the simulation performance and multi-omics application further indicate practical utility for high-dimensional integrative analyses.

major comments (2)

[§4] §4 (asymptotic theory): The posterior contraction and CLT statements are derived conditionally on the spectrally estimated factors. To transfer these guarantees to the unconditional procedure, uniform bounds on the spectral estimation error (in operator or Frobenius norm) that are o_p of the target contraction rate are required when both dimension p and the number of views grow; the manuscript does not appear to supply such bounds, leaving open whether the plug-in step preserves the stated rates or only approximates them.
[§3.2] §3.2 (posterior construction): The claim that the posterior is exactly a product of normal-inverse-gamma distributions relies on treating the spectral factor estimates as fixed. Because the factor estimates are data-dependent, the joint posterior for loadings and variances is no longer exactly conjugate with respect to the original data-generating measure; an explicit error-propagation argument or marginalization step would be needed to justify the exact factorization used for uncertainty quantification.

minor comments (2)

[§2] Notation for the number of views and the alignment step across views should be introduced earlier and used consistently in the model statement.
[Figure 1] Figure 1 (simulation results): axis labels and legend entries are too small for readability; consider increasing font size or splitting into two panels.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4 (asymptotic theory): The posterior contraction and CLT statements are derived conditionally on the spectrally estimated factors. To transfer these guarantees to the unconditional procedure, uniform bounds on the spectral estimation error (in operator or Frobenius norm) that are o_p of the target contraction rate are required when both dimension p and the number of views grow; the manuscript does not appear to supply such bounds, leaving open whether the plug-in step preserves the stated rates or only approximates them.

Authors: We agree with the referee that the current asymptotic results are conditional on the factor estimates obtained via spectral decomposition. In the revised manuscript, we will add a new subsection in §4 providing uniform bounds on the spectral estimation error in the operator norm. Under the assumptions of the factor model and with appropriate growth rates for p and the number of views, we show that this error is of smaller order than the posterior contraction rate, thereby transferring the guarantees to the unconditional procedure. This will be supported by additional lemmas on the concentration of the spectral estimators. revision: yes
Referee: [§3.2] §3.2 (posterior construction): The claim that the posterior is exactly a product of normal-inverse-gamma distributions relies on treating the spectral factor estimates as fixed. Because the factor estimates are data-dependent, the joint posterior for loadings and variances is no longer exactly conjugate with respect to the original data-generating measure; an explicit error-propagation argument or marginalization step would be needed to justify the exact factorization used for uncertainty quantification.

Authors: The referee correctly notes that the exact conjugacy and product-of-NIG form holds conditionally on the fixed factor estimates. We will revise the text in §3.2 to clarify this conditional nature explicitly. Additionally, we will include an error-propagation argument demonstrating that the total variation distance or other discrepancy between the conditional posterior and the true joint posterior vanishes asymptotically at the rates established in §4. This justifies the use of the conditional posterior for practical uncertainty quantification in high dimensions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; posterior follows from standard conjugate update conditional on external spectral estimates

full rationale

The derivation estimates latent factors via spectral decomposition of the observed data, then conditions on those point estimates to select jointly conjugate priors for loadings and variances. The resulting posterior is obtained by direct application of Bayes' rule under the conditional model, producing the claimed product of normal-inverse-gamma distributions. This is a standard, non-circular construction. The increasing-dimension posterior contraction and CLT results are established via separate asymptotic analysis under stated assumptions on the factor estimates; they do not reduce to redefinitions or self-citations of the target quantities. The method is self-contained against external benchmarks once the spectral step is accepted as a fixed input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard multi-view factor model assumption plus the technical premise that spectral methods recover the active factors sufficiently well for the conjugate posterior to be valid.

axioms (1)

domain assumption The observed multi-view data arise from a factor model containing both shared and view-specific latent factors.
This structure is invoked to justify the spectral decomposition step and the subsequent conjugate prior choice.

pith-pipeline@v0.9.0 · 5675 in / 1302 out tokens · 56139 ms · 2026-05-18T19:03:10.421864+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

employs spectral decompositions to estimate and align latent factors... resulting posterior is a simple product of normal-inverse gamma distributions... prove favorable increasing-dimension asymptotic properties, including posterior contraction and central limit theorems
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

JIC-based selection of latent dimensions and variance-inflation tuning for frequentist coverage

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Akbani, R., Ng, P. K. S., Werner, H. M. J. & et al. (2014), ‘A pan-cancer proteomic perspective on The Cancer Genome Atlas’,Nature Communications 5,

work page 2014
[2]

& Piferrer, F

URL: https://doi.org/10.1038/ncomms4887 Anastasiadi, D., Esteve-Codina, A. & Piferrer, F. (2018), ‘Consistent inverse correlation between DNA methylation of the first intron and gene expression across tissues and species’,Epigenetics & Chromatin 11(1),

work page doi:10.1038/ncomms4887 2018
[3]

URL: https://doi.org/10.1186/s13072-018-0205-1 Anceschi, N., Ferrari, F., Dunson, D. B. & Mallick, H. (2024), ‘Bayesian joint additive factor models for multiview learning’,arXiv preprint arXiv:2406.00778 . Argelaguet, R., Arnol, D., Bredikhin, D., Deloro, Y., Velten, B., Marioni, J. & Stegle, O. (2020), ‘MOFA+: A statistical framework for comprehensive i...

work page doi:10.1186/s13072-018-0205-1 2024
[4]

C., Buettner, F., Huber, W

Argelaguet, R., Velten, B., Arnol, D., Dietrich, S., Zenz, T., Marioni, J. C., Buettner, F., Huber, W. & Stegle, O. (2018), ‘Multi-omics factor analysis—a framework for unsupervised integration of multi-omics data sets’,Molecular Systems Biology 14(6), e8124. Bai, J. (2003), ‘Inferential theory for factor models of large dimensions’,Econometrica 71(1), 135–

work page 2018
[5]

18 Bandeira, A. S. & van Handel, R. (2016), ‘Sharp nonasymptotic bounds on the norm of random matrices with independent entries’,The Annals of Probability 44(4), 2479 –

work page 2016
[6]

D., Urz´ ua-Traslavi˜na, C

Bhattacharya, A., Bense, R. D., Urz´ ua-Traslavi˜na, C. G. et al. (2020), ‘Transcriptional effects of copy number alterations in a large set of human cancers’,Nature Communications 11(1),

work page 2020
[7]

URL: https://doi.org/10.1038/s41467-020-14605-5 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. (2021), ‘Determining the number of factors in high-dimensional generalized latent factor models’,Biometrika 109(3), 769–782. Chen, Y...

work page doi:10.1038/s41467-020-14605-5 2024
[8]

& Tshimanga-Ilunga, J

Gratton, S. & Tshimanga-Ilunga, J. (2016), ‘On a second-order expansion of the truncated singular subspace decomposition’,Numerical Linear Algebra with Applications 23(3), 519–534. 19 Gry, M., Rimini, R., Str ¨omberg, S., Asplund, A., Pont ´en, F., Uhlen, M. & Nilsson, P. (2009), ‘Correlations between RNA and protein expression profiles in 23 human cell l...

work page 2016
[9]

(1976), Modern Factor Analysis, University of Chicago Press

Harman, H. (1976), Modern Factor Analysis, University of Chicago Press. Hoerl, A. E. & Kennard, R. W. (1970), ‘Ridge Regression: Biased Estimation for Nonorthogonal Problems’,Technometrics 12(1), 55–67. Hutchinson, L. (2017), ‘Aneuploidy and immune evasion — a biomarker of response’, Nature Reviews Clinical Oncology 14(3),

work page 1976
[10]

URL: https://doi.org/10.1038/nrclinonc.2017.23 Ishwaran, B. H., J. & Rao, S. (2005), ‘Spike and slab variable selection: Frequentist and bayesian strategies’,The Annals of Statistics 33, 730–773. URL: https://api.semanticscholar.org/CorpusID:9004248 Jia, G., Ramalingam, T. R., Heiden, J. V., Gao, X., DePianto, D., Morshead, K. B., Modrusan, Z., Ramamoorth...

work page doi:10.1038/nrclinonc.2017.23 2017
[11]

M., Chen, Y

Lee, S. M., Chen, Y. & Sit, T. (2024), ‘A latent variable approach to learning high-dimensional multivariate longitudinal data’,arXiv preprint arXiv:2405.15053 . Lock, E. F., Hoadley, K. A., Marron, J. S. & Nobel, A. B. (2013), ‘Joint and individual variation explained (JIVE) for integrated analysis of multiple data types’,The Annals of Applied Statistics...

work page arXiv 2024
[12]

Y., Tang, T

Loh, C.-Y., Chai, J. Y., Tang, T. F., Wong, W. F., Sethi, G., Shanmugam, M. K., Chong, P. P. & Looi, C. Y. (2019), ‘The E-cadherin and N-cadherin switch in epithelial-to-mesenchymal transition: Signaling, therapeutic implications, and challenges’,Cells 8(10),

work page 2019
[13]

Ma, Z. & Ma, R. (2024), ‘Optimal estimation of shared singular subspaces across multiple noisy matrices’,arXiv preprint arXiv:2411.17054 . 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. (2025), ‘Factor pre-training in Bayesian multiva...

work page doi:10.3389/fonc.2020.550185 2024
[14]

Inference on covariance structure in high- dimensional multi-view data

Zito, A. & Miller, J. W. (2024), ‘Compressive Bayesian non-negative matrix factorization for mutational signatures analysis’,arXiv preprint arXiv:2404.10974 . 21 Supplementary material for “Inference on covariance structure in high- dimensional multi-view data” A Preliminary results Proposition 1 (Adapted from Proposition 3.5 of Chattopadhyay et al. (2024...

work page arXiv 2024
[15]

The result follows from Proposition 3.5 of Chattopadhyay et al

Since 𝑈0𝑚 = ˜𝑈0𝑚𝑅 for some orthogonal matrix 𝑅, where ˜𝑈0𝑚 is the matrix of left singular vectors of 𝐹0𝑚Λ⊤ 0𝑚, we have 𝑈0𝑚𝑈⊤ 0𝑚 = ˜𝑈0𝑚 ˜𝑈⊤ 0𝑚. The result follows from Proposition 3.5 of Chattopadhyay et al. (2024). Proposition

work page 2024
[16]

22 Therefore, denoting with 𝐸 = 1 𝑀 Í𝑀 𝑚=1 Δ𝑚, we have || 𝐸 || < 1/𝑀 with probability at least 1 − 𝑜(1), for 𝑛 and 𝑝min sufficiently large

Combining all of the above, we obtain 𝑠𝑘0 1 𝑀 𝑀∑︁ 𝑚=1 𝑃0𝑚 ≥ 1 𝑀 𝑠𝑘0 (𝐷0)2𝑑 −2 ≍ 1 𝑀 , with probability at least 1 − 𝑜(1), since by Corollary 5.35 in Vershynin (2012), we have 𝑠𝑘0 (𝐷0) =√𝑛 − O (𝑘0) and 𝑑 = √𝑛 + O (Í 𝑚 𝑘 𝑚) with probability at least 1 − 𝑜(1). 22 Therefore, denoting with 𝐸 = 1 𝑀 Í𝑀 𝑚=1 Δ𝑚, we have || 𝐸 || < 1/𝑀 with probability at least 1 − ...

work page 2012
[17]

we have min𝑅:𝑅⊤ 𝑅=𝐼𝑘0 ||𝑈 −𝑈0𝑅|| = ||𝑈 −𝑈0 ˆ𝑅|| ≲ 1 𝑛 + 1 𝑝min , with probability at least 1 − 𝑜(1), where ˆ𝑅 is the orthogonal matrix achieving the minimum of the quantity on the left hand side. Recalling that ˆ𝐹 = √𝑛𝑈 and letting ˜𝑅 = 𝑉0 ˆ𝑅, we have || ˆ𝐹 − 𝐹0 ˜𝑅|| = || √𝑛𝑈 − 𝑈0𝐷0 ˆ𝑅|| ≤ || √𝑛(𝑈 − 𝑈0 ˆ𝑅)|| + || √𝑛𝑈0 ˆ𝑅 − 𝑈0𝐷0 ˆ𝑅|| ≤ √𝑛 1 𝑛 + 1 𝑝min + ma...

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[18]

Recall thatˆΛ𝑚 = 𝑌 ⊤ 𝑚𝑈√𝑛/(𝑛+ 𝜏−2 𝑚 )

We start by establishing consistency of point estimates. Recall thatˆΛ𝑚 = 𝑌 ⊤ 𝑚𝑈√𝑛/(𝑛+ 𝜏−2 𝑚 ). Let 𝑚, 𝑚′ ∈ { 1, . . . , 𝑀 }, not necessarily distinct, and 𝑔𝑚𝑚′ = 𝑛 (𝑛+𝜏 −2𝑚 ) (𝑛+𝜏 −2 𝑚′ ) then ˆΛ𝑚 ˆΛ⊤ 𝑚′ = 𝑔𝑚𝑚′𝑌 ⊤ 𝑚𝑈𝑈 ⊤𝑌𝑚′ = 𝑔𝑚𝑚′𝑌 ⊤ 𝑚𝑈0𝑈⊤ 0 𝑌𝑚′ + 𝑔𝑚𝑚′𝑌 ⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑌𝑚′ and 𝑛 (𝑛 + 𝜏−2𝑚 )2 𝑌𝑚𝑈0𝑈⊤ 0 𝑌𝑚′ = 𝑛 (𝑛 + 𝜏−2𝑚 )2 𝐹0 𝐴𝑚Λ⊤ 0𝑚 + 𝐸𝑚 ⊤𝑈0𝑈⊤ 0 𝐹0 𝐴𝑚′ Λ...

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[19]

In addition, with probability at least 1 − 𝑜(1), we have || 𝐸 ⊤ 𝑚𝐹0𝑚′ Λ⊤ 0𝑚′ || ≲ (√𝑛 + √𝑝𝑚)√𝑛𝑝 𝑚′ , || 𝐸 ⊤ 𝑚′ 𝐹0𝑚Λ⊤ 0𝑚|| ≲ (√𝑛 + √𝑝𝑚′ )√𝑛𝑝 𝑚 and || 𝐸 ⊤ 𝑚𝑈0𝑈⊤ 0 𝐸𝑚′ || ≲ (√𝑛 + √𝑝𝑚) (√𝑛 + √𝑝𝑚′ ) since || 𝐸𝑚|| ≲ √𝑛 + √𝑝𝑚, || 𝐸𝑚′ || ≲ √𝑛 + √𝑝𝑚′ || 𝐹0𝑚|| ≍ || 𝐹0𝑚′ || ≲ √𝑛, with probability at least 1 − 𝑜(1), by Lemma 4 and Corollary 5.35 of Vershynin (2012), ...

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[20]

Thus, ˆ𝜆⊤ 𝑚 𝑗 ˆ𝜆𝑚′ 𝑗′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤𝑦 ( 𝑗′ ) 𝑚′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ + 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′

Recall ˆ𝜆𝑙 𝑗 = √𝑛 𝑛+𝜏 −2 𝑙 𝑈⊤𝑦 ( 𝑗 ) 𝑙 , where 𝑦 ( 𝑗 ) 𝑙 denotes the 𝑗-th column of 𝑌𝑙. Thus, ˆ𝜆⊤ 𝑚 𝑗 ˆ𝜆𝑚′ 𝑗′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤𝑦 ( 𝑗′ ) 𝑚′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ + 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ . where 𝑔𝑚𝑚′ = 𝑛 (𝑛+𝜏 −2𝑚 ) (𝑛+𝜏 −2 𝑚′ ) and we note 𝑔𝑚𝑚′ = 1 𝑛 + ℎ𝑚𝑚′, with ℎ𝑚𝑚′ ≍ 𝑂 (1/𝑛2). We decompose the terms: 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈...

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[21]

First, we focus on the case 𝑚 = 𝑚′. In particular, by the central limit theorem, we have √𝑛 1 𝑛 𝜆⊤ 0𝑚 𝑗 𝐴⊤ 𝑚𝐹⊤ 0 𝐹0 𝐴𝑚𝜆0𝑚 𝑗′ − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ = √𝑛𝜆⊤ 0𝑚 𝑗 1 𝑛 𝐹⊤ 0𝑚𝐹0𝑚𝜆0𝑚 𝑗′ − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ = √𝑛 1 𝑛 𝑛∑︁ 𝑖=1 (𝜂⊤ 0𝑚𝑖𝜆0𝑚 𝑗) (𝜂⊤ 0𝑚𝑖𝜆0𝑚 𝑗′ ) − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ ⇒ 𝑁 (0, Ω2 0𝑚 𝑗 𝑗′ ), 25 where Ω2 0𝑚 𝑗 𝑗′ = ( (𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ )2 + || 𝜆0𝑚 𝑗 || 2|| 𝜆0𝑚 𝑗′ || 2, if 𝑗 ≠ 𝑗 ′,...

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[22]

We follows similar steps as in Theorem 3.7 of Chattopadhyay et al. (2024). Recall that a sample for˜𝜆𝑚𝑙 from ˜Π can be represented as˜𝜆𝑚𝑙 = ˆ𝜆𝑚𝑙 + 𝜌𝑚 ˜𝜎𝑚 𝑗√ 𝑛+𝜏 −2𝑚 ˜𝑒𝑚 𝑗, where ˜𝑒𝑚 𝑗 ∼ 𝑁𝑘0 (0, 𝐼𝑘0), ˜𝜎𝑚 𝑗 ∼ 𝐼𝐺 (𝜈𝑛/2𝜈𝑛𝛿2 𝑚 𝑗/2). Thus, ˜𝜆⊤ 𝑚 𝑗 ˜𝜆𝑚′ 𝑗′ = ˆ𝜆⊤ 𝑚𝑙 ˆ𝜆𝑚′ 𝑗′ + ˜𝜎𝑚′ 𝑗′ 𝜌𝑚′ √︃ 𝑛 + 𝜏−2 𝑚′ ˆ𝜆⊤ 𝑚𝑙 ˜𝑒𝑚′ 𝑗′ + ˜𝜎𝑚 𝑗 𝜌𝑚 √︁ 𝑛 + 𝜏−2𝑚 ˆ𝜆⊤ 𝑚′𝑙′ ˜𝑒𝑚 𝑗 + ˜𝜎𝑚 𝑗 ...

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[23]

(2024) concludes the proof

An application of Lemma F.2 of Chattopadhyay et al. (2024) concludes the proof. C Additional results Proposition

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[24]

The first result follows from max𝑗=1,..., 𝑝 𝑚 || 𝑒 ( 𝑗 ) 𝑚 || ≲ √𝑛 Lemma 1 of Laurent & Massart (2000) combined with Assumption

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[25]

Recall that, with probability at least 1 − 𝑜(1), || 𝐹0𝑚|| ≍ √𝑛 by Corollary 5.35 of Vershynin (2012) and max𝑗=1,..., 𝑝 𝑚 || 𝜆0𝑚 𝑗 || ≤ || Λ0||∞ √𝑘 𝑚 ≍ 1 by Assumption

Next, consider the following || 𝑦 ( 𝑗 ) 𝑚 || ≤ || 𝐹0𝑚|||| 𝜆0𝑚 𝑗 || + || 𝑒 ( 𝑗 ) 𝑚 || . Recall that, with probability at least 1 − 𝑜(1), || 𝐹0𝑚|| ≍ √𝑛 by Corollary 5.35 of Vershynin (2012) and max𝑗=1,..., 𝑝 𝑚 || 𝜆0𝑚 𝑗 || ≤ || Λ0||∞ √𝑘 𝑚 ≍ 1 by Assumption

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[26]

Then, by Corollary 3.11 in Bandeira & van Handel (2016), we have || 𝐸𝑚|| ≲ √𝑛𝜎𝑚,𝑚𝑎𝑥 + 𝜎𝑚,𝑠𝑢𝑚 ≤ 𝜎𝑚,𝑚𝑎𝑥 (√𝑛 + √𝑝𝑚), with probability at least 1 − 𝑜(1)

Let 𝜎2 𝑚,𝑚𝑎𝑥 = max𝑗=1,..., 𝑝 𝑚 𝜎2 0𝑚 𝑗 and 𝜎2 𝑚,𝑠𝑢𝑚 = Í 𝑝𝑚 𝑗=1 𝜎2 0𝑚 𝑗 ≤ 𝑝𝑚𝜎2 𝑚,𝑚𝑎𝑥 . Then, by Corollary 3.11 in Bandeira & van Handel (2016), we have || 𝐸𝑚|| ≲ √𝑛𝜎𝑚,𝑚𝑎𝑥 + 𝜎𝑚,𝑠𝑢𝑚 ≤ 𝜎𝑚,𝑚𝑎𝑥 (√𝑛 + √𝑝𝑚), with probability at least 1 − 𝑜(1). The first result follows from Assumption

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[27]

For the second result, note that || 𝐹𝑚Λ⊤ 0𝑚|| ≲ √𝑛𝑝 𝑚 with probability at least 1 − 𝑜(1), since || Λ0𝑚|| ≍ √𝑝𝑚 by Assumption 4 and || 𝐹𝑚|| ≍ √𝑛 with probability at least 1 − 𝑜(1) by Corollary 5.35 in Vershynin (2012). Lemma

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[28]

We follow similar steps of the proof of part (b) of Theorem 3.6 in Chattopadhyay et al. (2024). Consider the following decomposition ˜𝜎2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 = 1 − 2𝑈𝑚 𝑗 𝜈𝑛 𝛿2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 − 2𝑈𝑗 𝜈𝑛 𝜎2 0 𝑗 , where 𝑈𝑚 𝑗 = 𝜈𝑛 2 𝛿2 𝑚 𝑗 ˜𝜎−2 𝑚 𝑗 − 1 . With probability at least 1− 𝑜(1), we have max𝑗=1,..., 𝑝 𝑚 |𝑈𝑚 𝑗 |/𝛾𝑛 ≲ (log 𝑝𝑚/𝑛)1/3, by Lemma E.7 in Chattopadhyay...

work page 2024
[29]

Moreover, with probability at least 1 − 𝑜(1), max 𝑗=1,..., 𝑝 𝑚 𝐶𝑗 𝜎2 0𝑚 𝑗 − ( 𝑛 − 𝑘 𝑚) ≲ 𝑛 log 𝑝𝑚 𝑛 1/3, by Lemma E.7 of Chattopadhyay et al. (2024). Combining all of the above, we get max 𝑗=1,..., 𝑝 𝑚 | ˜𝜎2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 | ≲ log 𝑝𝑚 𝑛 1/3 + 1 𝑝min , with probability at least 1 − 𝑜(1). 31 E Additional details about the selection of the number of latent fac...

work page 2024

[1] [1]

Akbani, R., Ng, P. K. S., Werner, H. M. J. & et al. (2014), ‘A pan-cancer proteomic perspective on The Cancer Genome Atlas’,Nature Communications 5,

work page 2014

[2] [2]

& Piferrer, F

URL: https://doi.org/10.1038/ncomms4887 Anastasiadi, D., Esteve-Codina, A. & Piferrer, F. (2018), ‘Consistent inverse correlation between DNA methylation of the first intron and gene expression across tissues and species’,Epigenetics & Chromatin 11(1),

work page doi:10.1038/ncomms4887 2018

[3] [3]

URL: https://doi.org/10.1186/s13072-018-0205-1 Anceschi, N., Ferrari, F., Dunson, D. B. & Mallick, H. (2024), ‘Bayesian joint additive factor models for multiview learning’,arXiv preprint arXiv:2406.00778 . Argelaguet, R., Arnol, D., Bredikhin, D., Deloro, Y., Velten, B., Marioni, J. & Stegle, O. (2020), ‘MOFA+: A statistical framework for comprehensive i...

work page doi:10.1186/s13072-018-0205-1 2024

[4] [4]

C., Buettner, F., Huber, W

Argelaguet, R., Velten, B., Arnol, D., Dietrich, S., Zenz, T., Marioni, J. C., Buettner, F., Huber, W. & Stegle, O. (2018), ‘Multi-omics factor analysis—a framework for unsupervised integration of multi-omics data sets’,Molecular Systems Biology 14(6), e8124. Bai, J. (2003), ‘Inferential theory for factor models of large dimensions’,Econometrica 71(1), 135–

work page 2018

[5] [5]

18 Bandeira, A. S. & van Handel, R. (2016), ‘Sharp nonasymptotic bounds on the norm of random matrices with independent entries’,The Annals of Probability 44(4), 2479 –

work page 2016

[6] [6]

D., Urz´ ua-Traslavi˜na, C

Bhattacharya, A., Bense, R. D., Urz´ ua-Traslavi˜na, C. G. et al. (2020), ‘Transcriptional effects of copy number alterations in a large set of human cancers’,Nature Communications 11(1),

work page 2020

[7] [7]

URL: https://doi.org/10.1038/s41467-020-14605-5 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. (2021), ‘Determining the number of factors in high-dimensional generalized latent factor models’,Biometrika 109(3), 769–782. Chen, Y...

work page doi:10.1038/s41467-020-14605-5 2024

[8] [8]

& Tshimanga-Ilunga, J

Gratton, S. & Tshimanga-Ilunga, J. (2016), ‘On a second-order expansion of the truncated singular subspace decomposition’,Numerical Linear Algebra with Applications 23(3), 519–534. 19 Gry, M., Rimini, R., Str ¨omberg, S., Asplund, A., Pont ´en, F., Uhlen, M. & Nilsson, P. (2009), ‘Correlations between RNA and protein expression profiles in 23 human cell l...

work page 2016

[9] [9]

(1976), Modern Factor Analysis, University of Chicago Press

Harman, H. (1976), Modern Factor Analysis, University of Chicago Press. Hoerl, A. E. & Kennard, R. W. (1970), ‘Ridge Regression: Biased Estimation for Nonorthogonal Problems’,Technometrics 12(1), 55–67. Hutchinson, L. (2017), ‘Aneuploidy and immune evasion — a biomarker of response’, Nature Reviews Clinical Oncology 14(3),

work page 1976

[10] [10]

URL: https://doi.org/10.1038/nrclinonc.2017.23 Ishwaran, B. H., J. & Rao, S. (2005), ‘Spike and slab variable selection: Frequentist and bayesian strategies’,The Annals of Statistics 33, 730–773. URL: https://api.semanticscholar.org/CorpusID:9004248 Jia, G., Ramalingam, T. R., Heiden, J. V., Gao, X., DePianto, D., Morshead, K. B., Modrusan, Z., Ramamoorth...

work page doi:10.1038/nrclinonc.2017.23 2017

[11] [11]

M., Chen, Y

Lee, S. M., Chen, Y. & Sit, T. (2024), ‘A latent variable approach to learning high-dimensional multivariate longitudinal data’,arXiv preprint arXiv:2405.15053 . Lock, E. F., Hoadley, K. A., Marron, J. S. & Nobel, A. B. (2013), ‘Joint and individual variation explained (JIVE) for integrated analysis of multiple data types’,The Annals of Applied Statistics...

work page arXiv 2024

[12] [12]

Y., Tang, T

Loh, C.-Y., Chai, J. Y., Tang, T. F., Wong, W. F., Sethi, G., Shanmugam, M. K., Chong, P. P. & Looi, C. Y. (2019), ‘The E-cadherin and N-cadherin switch in epithelial-to-mesenchymal transition: Signaling, therapeutic implications, and challenges’,Cells 8(10),

work page 2019

[13] [13]

Ma, Z. & Ma, R. (2024), ‘Optimal estimation of shared singular subspaces across multiple noisy matrices’,arXiv preprint arXiv:2411.17054 . 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. (2025), ‘Factor pre-training in Bayesian multiva...

work page doi:10.3389/fonc.2020.550185 2024

[14] [14]

Inference on covariance structure in high- dimensional multi-view data

Zito, A. & Miller, J. W. (2024), ‘Compressive Bayesian non-negative matrix factorization for mutational signatures analysis’,arXiv preprint arXiv:2404.10974 . 21 Supplementary material for “Inference on covariance structure in high- dimensional multi-view data” A Preliminary results Proposition 1 (Adapted from Proposition 3.5 of Chattopadhyay et al. (2024...

work page arXiv 2024

[15] [15]

The result follows from Proposition 3.5 of Chattopadhyay et al

Since 𝑈0𝑚 = ˜𝑈0𝑚𝑅 for some orthogonal matrix 𝑅, where ˜𝑈0𝑚 is the matrix of left singular vectors of 𝐹0𝑚Λ⊤ 0𝑚, we have 𝑈0𝑚𝑈⊤ 0𝑚 = ˜𝑈0𝑚 ˜𝑈⊤ 0𝑚. The result follows from Proposition 3.5 of Chattopadhyay et al. (2024). Proposition

work page 2024

[16] [16]

22 Therefore, denoting with 𝐸 = 1 𝑀 Í𝑀 𝑚=1 Δ𝑚, we have || 𝐸 || < 1/𝑀 with probability at least 1 − 𝑜(1), for 𝑛 and 𝑝min sufficiently large

Combining all of the above, we obtain 𝑠𝑘0 1 𝑀 𝑀∑︁ 𝑚=1 𝑃0𝑚 ≥ 1 𝑀 𝑠𝑘0 (𝐷0)2𝑑 −2 ≍ 1 𝑀 , with probability at least 1 − 𝑜(1), since by Corollary 5.35 in Vershynin (2012), we have 𝑠𝑘0 (𝐷0) =√𝑛 − O (𝑘0) and 𝑑 = √𝑛 + O (Í 𝑚 𝑘 𝑚) with probability at least 1 − 𝑜(1). 22 Therefore, denoting with 𝐸 = 1 𝑀 Í𝑀 𝑚=1 Δ𝑚, we have || 𝐸 || < 1/𝑀 with probability at least 1 − ...

work page 2012

[17] [17]

we have min𝑅:𝑅⊤ 𝑅=𝐼𝑘0 ||𝑈 −𝑈0𝑅|| = ||𝑈 −𝑈0 ˆ𝑅|| ≲ 1 𝑛 + 1 𝑝min , with probability at least 1 − 𝑜(1), where ˆ𝑅 is the orthogonal matrix achieving the minimum of the quantity on the left hand side. Recalling that ˆ𝐹 = √𝑛𝑈 and letting ˜𝑅 = 𝑉0 ˆ𝑅, we have || ˆ𝐹 − 𝐹0 ˜𝑅|| = || √𝑛𝑈 − 𝑈0𝐷0 ˆ𝑅|| ≤ || √𝑛(𝑈 − 𝑈0 ˆ𝑅)|| + || √𝑛𝑈0 ˆ𝑅 − 𝑈0𝐷0 ˆ𝑅|| ≤ √𝑛 1 𝑛 + 1 𝑝min + ma...

work page 2012

[18] [18]

Recall thatˆΛ𝑚 = 𝑌 ⊤ 𝑚𝑈√𝑛/(𝑛+ 𝜏−2 𝑚 )

We start by establishing consistency of point estimates. Recall thatˆΛ𝑚 = 𝑌 ⊤ 𝑚𝑈√𝑛/(𝑛+ 𝜏−2 𝑚 ). Let 𝑚, 𝑚′ ∈ { 1, . . . , 𝑀 }, not necessarily distinct, and 𝑔𝑚𝑚′ = 𝑛 (𝑛+𝜏 −2𝑚 ) (𝑛+𝜏 −2 𝑚′ ) then ˆΛ𝑚 ˆΛ⊤ 𝑚′ = 𝑔𝑚𝑚′𝑌 ⊤ 𝑚𝑈𝑈 ⊤𝑌𝑚′ = 𝑔𝑚𝑚′𝑌 ⊤ 𝑚𝑈0𝑈⊤ 0 𝑌𝑚′ + 𝑔𝑚𝑚′𝑌 ⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑌𝑚′ and 𝑛 (𝑛 + 𝜏−2𝑚 )2 𝑌𝑚𝑈0𝑈⊤ 0 𝑌𝑚′ = 𝑛 (𝑛 + 𝜏−2𝑚 )2 𝐹0 𝐴𝑚Λ⊤ 0𝑚 + 𝐸𝑚 ⊤𝑈0𝑈⊤ 0 𝐹0 𝐴𝑚′ Λ...

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[19] [19]

In addition, with probability at least 1 − 𝑜(1), we have || 𝐸 ⊤ 𝑚𝐹0𝑚′ Λ⊤ 0𝑚′ || ≲ (√𝑛 + √𝑝𝑚)√𝑛𝑝 𝑚′ , || 𝐸 ⊤ 𝑚′ 𝐹0𝑚Λ⊤ 0𝑚|| ≲ (√𝑛 + √𝑝𝑚′ )√𝑛𝑝 𝑚 and || 𝐸 ⊤ 𝑚𝑈0𝑈⊤ 0 𝐸𝑚′ || ≲ (√𝑛 + √𝑝𝑚) (√𝑛 + √𝑝𝑚′ ) since || 𝐸𝑚|| ≲ √𝑛 + √𝑝𝑚, || 𝐸𝑚′ || ≲ √𝑛 + √𝑝𝑚′ || 𝐹0𝑚|| ≍ || 𝐹0𝑚′ || ≲ √𝑛, with probability at least 1 − 𝑜(1), by Lemma 4 and Corollary 5.35 of Vershynin (2012), ...

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[20] [20]

Thus, ˆ𝜆⊤ 𝑚 𝑗 ˆ𝜆𝑚′ 𝑗′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤𝑦 ( 𝑗′ ) 𝑚′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ + 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′

Recall ˆ𝜆𝑙 𝑗 = √𝑛 𝑛+𝜏 −2 𝑙 𝑈⊤𝑦 ( 𝑗 ) 𝑙 , where 𝑦 ( 𝑗 ) 𝑙 denotes the 𝑗-th column of 𝑌𝑙. Thus, ˆ𝜆⊤ 𝑚 𝑗 ˆ𝜆𝑚′ 𝑗′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤𝑦 ( 𝑗′ ) 𝑚′ = 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ + 𝑔𝑚𝑚′ 𝑦 ( 𝑗 )⊤ 𝑚 𝑈𝑈 ⊤ − 𝑈0𝑈⊤ 0 𝑦 ( 𝑗′ ) 𝑚′ . where 𝑔𝑚𝑚′ = 𝑛 (𝑛+𝜏 −2𝑚 ) (𝑛+𝜏 −2 𝑚′ ) and we note 𝑔𝑚𝑚′ = 1 𝑛 + ℎ𝑚𝑚′, with ℎ𝑚𝑚′ ≍ 𝑂 (1/𝑛2). We decompose the terms: 𝑦 ( 𝑗 )⊤ 𝑚 𝑈0𝑈...

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[21] [21]

First, we focus on the case 𝑚 = 𝑚′. In particular, by the central limit theorem, we have √𝑛 1 𝑛 𝜆⊤ 0𝑚 𝑗 𝐴⊤ 𝑚𝐹⊤ 0 𝐹0 𝐴𝑚𝜆0𝑚 𝑗′ − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ = √𝑛𝜆⊤ 0𝑚 𝑗 1 𝑛 𝐹⊤ 0𝑚𝐹0𝑚𝜆0𝑚 𝑗′ − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ = √𝑛 1 𝑛 𝑛∑︁ 𝑖=1 (𝜂⊤ 0𝑚𝑖𝜆0𝑚 𝑗) (𝜂⊤ 0𝑚𝑖𝜆0𝑚 𝑗′ ) − 𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ ⇒ 𝑁 (0, Ω2 0𝑚 𝑗 𝑗′ ), 25 where Ω2 0𝑚 𝑗 𝑗′ = ( (𝜆⊤ 0𝑚 𝑗𝜆0𝑚 𝑗′ )2 + || 𝜆0𝑚 𝑗 || 2|| 𝜆0𝑚 𝑗′ || 2, if 𝑗 ≠ 𝑗 ′,...

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[22] [22]

We follows similar steps as in Theorem 3.7 of Chattopadhyay et al. (2024). Recall that a sample for˜𝜆𝑚𝑙 from ˜Π can be represented as˜𝜆𝑚𝑙 = ˆ𝜆𝑚𝑙 + 𝜌𝑚 ˜𝜎𝑚 𝑗√ 𝑛+𝜏 −2𝑚 ˜𝑒𝑚 𝑗, where ˜𝑒𝑚 𝑗 ∼ 𝑁𝑘0 (0, 𝐼𝑘0), ˜𝜎𝑚 𝑗 ∼ 𝐼𝐺 (𝜈𝑛/2𝜈𝑛𝛿2 𝑚 𝑗/2). Thus, ˜𝜆⊤ 𝑚 𝑗 ˜𝜆𝑚′ 𝑗′ = ˆ𝜆⊤ 𝑚𝑙 ˆ𝜆𝑚′ 𝑗′ + ˜𝜎𝑚′ 𝑗′ 𝜌𝑚′ √︃ 𝑛 + 𝜏−2 𝑚′ ˆ𝜆⊤ 𝑚𝑙 ˜𝑒𝑚′ 𝑗′ + ˜𝜎𝑚 𝑗 𝜌𝑚 √︁ 𝑛 + 𝜏−2𝑚 ˆ𝜆⊤ 𝑚′𝑙′ ˜𝑒𝑚 𝑗 + ˜𝜎𝑚 𝑗 ...

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[23] [23]

(2024) concludes the proof

An application of Lemma F.2 of Chattopadhyay et al. (2024) concludes the proof. C Additional results Proposition

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[24] [24]

The first result follows from max𝑗=1,..., 𝑝 𝑚 || 𝑒 ( 𝑗 ) 𝑚 || ≲ √𝑛 Lemma 1 of Laurent & Massart (2000) combined with Assumption

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[25] [25]

Recall that, with probability at least 1 − 𝑜(1), || 𝐹0𝑚|| ≍ √𝑛 by Corollary 5.35 of Vershynin (2012) and max𝑗=1,..., 𝑝 𝑚 || 𝜆0𝑚 𝑗 || ≤ || Λ0||∞ √𝑘 𝑚 ≍ 1 by Assumption

Next, consider the following || 𝑦 ( 𝑗 ) 𝑚 || ≤ || 𝐹0𝑚|||| 𝜆0𝑚 𝑗 || + || 𝑒 ( 𝑗 ) 𝑚 || . Recall that, with probability at least 1 − 𝑜(1), || 𝐹0𝑚|| ≍ √𝑛 by Corollary 5.35 of Vershynin (2012) and max𝑗=1,..., 𝑝 𝑚 || 𝜆0𝑚 𝑗 || ≤ || Λ0||∞ √𝑘 𝑚 ≍ 1 by Assumption

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[26] [26]

Then, by Corollary 3.11 in Bandeira & van Handel (2016), we have || 𝐸𝑚|| ≲ √𝑛𝜎𝑚,𝑚𝑎𝑥 + 𝜎𝑚,𝑠𝑢𝑚 ≤ 𝜎𝑚,𝑚𝑎𝑥 (√𝑛 + √𝑝𝑚), with probability at least 1 − 𝑜(1)

Let 𝜎2 𝑚,𝑚𝑎𝑥 = max𝑗=1,..., 𝑝 𝑚 𝜎2 0𝑚 𝑗 and 𝜎2 𝑚,𝑠𝑢𝑚 = Í 𝑝𝑚 𝑗=1 𝜎2 0𝑚 𝑗 ≤ 𝑝𝑚𝜎2 𝑚,𝑚𝑎𝑥 . Then, by Corollary 3.11 in Bandeira & van Handel (2016), we have || 𝐸𝑚|| ≲ √𝑛𝜎𝑚,𝑚𝑎𝑥 + 𝜎𝑚,𝑠𝑢𝑚 ≤ 𝜎𝑚,𝑚𝑎𝑥 (√𝑛 + √𝑝𝑚), with probability at least 1 − 𝑜(1). The first result follows from Assumption

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[27] [27]

For the second result, note that || 𝐹𝑚Λ⊤ 0𝑚|| ≲ √𝑛𝑝 𝑚 with probability at least 1 − 𝑜(1), since || Λ0𝑚|| ≍ √𝑝𝑚 by Assumption 4 and || 𝐹𝑚|| ≍ √𝑛 with probability at least 1 − 𝑜(1) by Corollary 5.35 in Vershynin (2012). Lemma

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[28] [28]

We follow similar steps of the proof of part (b) of Theorem 3.6 in Chattopadhyay et al. (2024). Consider the following decomposition ˜𝜎2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 = 1 − 2𝑈𝑚 𝑗 𝜈𝑛 𝛿2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 − 2𝑈𝑗 𝜈𝑛 𝜎2 0 𝑗 , where 𝑈𝑚 𝑗 = 𝜈𝑛 2 𝛿2 𝑚 𝑗 ˜𝜎−2 𝑚 𝑗 − 1 . With probability at least 1− 𝑜(1), we have max𝑗=1,..., 𝑝 𝑚 |𝑈𝑚 𝑗 |/𝛾𝑛 ≲ (log 𝑝𝑚/𝑛)1/3, by Lemma E.7 in Chattopadhyay...

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[29] [29]

Moreover, with probability at least 1 − 𝑜(1), max 𝑗=1,..., 𝑝 𝑚 𝐶𝑗 𝜎2 0𝑚 𝑗 − ( 𝑛 − 𝑘 𝑚) ≲ 𝑛 log 𝑝𝑚 𝑛 1/3, by Lemma E.7 of Chattopadhyay et al. (2024). Combining all of the above, we get max 𝑗=1,..., 𝑝 𝑚 | ˜𝜎2 𝑚 𝑗 − 𝜎2 0𝑚 𝑗 | ≲ log 𝑝𝑚 𝑛 1/3 + 1 𝑝min , with probability at least 1 − 𝑜(1). 31 E Additional details about the selection of the number of latent fac...

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