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Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Stack the data before the SVD: optimally weighted Stack-SVD dominates.

desk verdict Genuinely new asymptotic results for SVD-Stack and a clean comparison with Stack-SVD; send it to a serious referee. read the letter →

arxiv 2507.22170 v1 pith:HLE4LOWY submitted 2025-07-29 stat.ML cs.LGmath.STstat.MEstat.TH

Tavor Z. Baharav , Phillip B. Nicol , Rafael A. Irizarry , Rong Ma This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.MEstat.TH MSC 15A1862H25
keywords dataintegrationrandommatrixtheoryspectralmethodsStack-SVDSVD-Stackoptimalweightingphasetransitionsharedsubspace
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

Which pipeline should a practitioner use to recover a shared low-dimensional structure from several noisy high-dimensional tables: concatenate the tables and run one SVD (Stack-SVD), or run an SVD per table and then combine the top singular vectors (SVD-Stack)? This paper answers the question in the proportional regime where table sizes and dimension grow together, using random matrix theory to derive the limiting squared cosine $|\langle \hat v, v\rangle|^2$ between the estimated and true shared vector. It shows that neither pipeline wins in all instances when unweighted, but once each table is scaled by its signal-to-noise-optimal weight, Stack-SVD dominates SVD-Stack, strictly so when signal strengths differ and at least two tables carry signal. The paper also gives exact detectability thresholds, extends the comparison to multiple shared components, and supplies a consistent estimator of the signal strengths, so the optimal weights can be implemented from data.

What carries the argument

The argument runs through two limiting objects. For SVD-Stack, the $M\times M$ matrix $A_\beta = \beta\beta^{\top} + \mathrm{diag}(1-\beta_1^2,\ldots,1-\beta_M^2)$ is the entrywise limit of the Gram matrix of the per-table singular-vector estimates; its leading eigenvector and eigenvalue directly give the performance, once a delocalization lemma shows $|\langle \hat v_i, \hat v_j\rangle|^2 \to \beta_i^2 \beta_j^2$. For Stack-SVD, the load-bearing object is the secular equation $\sum_i \theta_i^4(1-x)/(c_i + x\theta_i^2) = 1$, whose unique root in $(0,1)$ is the weighted performance; the weights $w_i^* \propto \theta_i/\sqrt{\theta_i^2+c_i}$ are found by reparametrizing the weighted covariance's spectrum to match the weighted-PCA optimization problem. The dominance proof evaluates this secular equation at $x = S/(S+1)$, the weighted SVD-Stack limit, and uses a termwise inequality to show the root must be at least as large.

What would settle it

Simulate two independent tables with $d$ large and with i.i.d. noise that satisfies the paper's moment condition but is concentrated on a few coordinates, then measure $|\langle \hat v_1, \hat v_2\rangle|^2$ over many replicates; if it does not concentrate at $\beta_1^2\beta_2^2$, Lemma 1 fails and the SVD-Stack limits, along with the claimed dominance of weighted Stack-SVD, would need revision.

Watch

Extended reading notes

Core claim

Under the model $X_i = \theta_i u_i v^{\top} + E_i$ with $n_i/d \to c_i$ and i.i.d. noise of variance $1/d$ and bounded fourth moment, the paper determines, in probability, the asymptotic performance $|\langle \hat v, v\rangle|^2$ of both estimators. For unweighted Stack-SVD it converges to $(\|\theta\|_2^4 - \|c\|_1)/(\|\theta\|_2^2(\|\theta\|_2^2+1))$ above the threshold $\|\theta\|_2^4 > \|c\|_1$. For unweighted SVD-Stack it converges to $(\beta^{\top} v_{\max}(A_\beta))^2/\lambda_{\max}(A_\beta)$, where $A_\beta = \beta\beta^{\top} + \mathrm{diag}(1-\beta_1^2,\ldots,1-\beta_M^2)$ and $\beta_i^2 = (\theta_i^4-c_i)/(\theta_i^4+\theta_i^2)$ when $\theta_i^4 \ge c_i$, with $\beta_i=0$ otherwise. With optimal weights the limits become cleaner: Stack-SVD weights $w_i^* \propto \theta_i/\sqrt{\theta_i^2+c_i}$ give the unique root $x\in(0,1)$ of $\sum_i \theta_i^4(1-x)/(c_i+x\theta_i^2)=1$ whenever $\sum_i \theta_i^4/c_i > 1$, while SVD-Stack weights $w_i^* = \theta_i\sqrt{\theta_i^2+1}/(\theta_i^2+c_i)$ give $S/(S+1)$ with $S=\sum_i \beta_i^2/(1-\beta_i^2)$. The paper's main theorem states that optimally weighted Stack-SVD dominates unweighted Stack-SVD and optimally weighted SVD-Stack, with strict improvement whenever the $\theta_i$ are not all equal and at least two of them are nonzero; the same comparison holds componentwise for rank-$r$ shared subspaces.

Load-bearing premise

The SVD-Stack formulas depend on the noise scrambling the residual parts of each estimated singular vector evenly across all perpendicular directions, so that estimates from different tables overlap only through the true signal.

Editorial extensions

If this is right

  • With optimal weights, Stack-SVD never underperforms optimally weighted SVD-Stack or unweighted Stack-SVD; the improvement over SVD-Stack is strict when at least two tables carry signal, and the improvement over unweighted Stack-SVD is strict when the $\theta_i$ are not all equal.
  • Method choice for unweighted pipelines depends on the instance: Stack-SVD pools many subthreshold tables into a detectable signal, while SVD-Stack avoids dilution when a few high-noise tables with large $c_i$ are present.
  • Optimal weighting lowers the detection threshold: weighted Stack-SVD detects when $\sum_i \theta_i^4/c_i > 1$, a condition implied by the unweighted threshold $(\sum_i\theta_i^2)^2/\sum_i c_i > 1$ and strictly weaker when the ratios $\theta_i^2/c_i$ vary.
  • There exist instance sequences where weighted Stack-SVD approaches perfect recovery while unweighted Stack-SVD, SVD-Stack, binary-weighted Stack-SVD, and weighted SVD-Stack all fall below detectability.
  • For rank-$r$ shared subspaces, the componentwise formulas give subspace error $\|V^{\top}\hat V\|_F^2 \to \sum_j \gamma_j$ for weighted Stack-SVD and $\to \sum_j S_j/(S_j+1)$ for weighted SVD-Stack.

Reading between the lines

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

  • A testable extension is to replace the i.i.d. noise assumption with sparsely corrupted or row-correlated noise; if SVD-Stack's residuals no longer delocalize, its $S/(S+1)$ formula should degrade measurably, while weighted Stack-SVD's secular-equation limit should survive.
  • The paper's plug-in estimator of $\theta_i$ could be turned into an online rule that reweights tables as more data arrive; the paper establishes consistency but does not analyze the finite-sample behavior of the estimated weights.
  • In federated settings where raw tables cannot be shared, this result implies a trade-off: SVD-Stack is the communication-friendly option but is dominated by weighted Stack-SVD, so communicating per-table statistics such as estimated $\beta_i$ may partially close the gap; this is an inference, not a claim of the paper.
  • For practitioners building large genomic atlases, the dominance result supports pooling data before inference, but only when signal strengths can be estimated; the paper's starkest prediction is that many individually undetectable tables can be aggregated by unweighted Stack-SVD into strong recovery.
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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

1 major / 4 minor

Summary. This paper studies two pipelines for estimating a shared rank-one (and, by extension, finite-rank) singular subspace across multiple noisy high-dimensional matrices: Stack-SVD (concatenate then SVD) and SVD-Stack (per-matrix SVD then aggregate). Under a proportional scaling regime with independent noise entries having zero mean, unit variance after scaling, and bounded fourth moment, the authors derive the limit in probability of the squared inner product between the estimated and true singular vector for both methods, including phase transitions and closed-form asymptotic performance expressions. They further derive optimal weighting schemes for both methods, show that optimally weighted Stack-SVD dominates optimally weighted SVD-Stack (and unweighted Stack-SVD), provide algorithms for estimating the signal strengths from data, extend to rank r under ordering assumptions, and support the theory with extensive simulations, including non-Gaussian noise, and a semi-synthetic single-cell RNA-seq experiment.

Significance. If correct, this is the first systematic random matrix treatment of SVD-Stack and a rigorous instance-dependent comparison of the two methods in the proportional regime. The explicit asymptotic formulas, weighting rules, and detectability thresholds are directly usable by practitioners, and the public code plus the non-Gaussian and semi-synthetic validations strengthen the paper's practical claims. The main theoretical result—dominance of optimally weighted Stack-SVD—is novel and goes beyond the minimax analyses available in the literature.

major comments (1)
  1. [Section 5, Theorem 3] The claimed strict improvement over unweighted Stack-SVD when the θ_i are not all equal is false. Take M=2, θ=(1,2), c=(1,4). Then θ_i^2/c_i = 1 for both tables, so the optimal weights of Theorem 1, w_i^* ∝ θ_i/√(θ_i^2 + c_i), are equal, and both unweighted and optimally weighted Stack-SVD have asymptotic performance 2/3, although the θ_i are not all equal. The correct condition for strict improvement is that the θ_i^2/c_i are not all equal (equivalently, that the optimal weights are not all proportional to the unweighted choice). The proof in Appendix C.2 establishes only the inequality f(S/(S+1)) ≥ 1 and does not support the strictness over unweighted Stack-SVD; the statement of Theorem 3 and the sketch in Section 5 should be amended accordingly.
minor comments (4)
  1. [Throughout] There are spacing typos such as 'SVD-Stackfalls' in the discussion of Figure 4a and 'V ariance' in Section 7; these should be corrected.
  2. [Section 8] The variance-stabilizing transformation is written as X = 2√Y/d; since Y is a count matrix, this is dimensionally ambiguous. Please clarify whether the intended transformation is X = 2√(Y/d) or X = 2√Y/√d, and adjust the stated variance 1/d accordingly.
  3. [Table 1] The detectability condition for weighted SVD-Stack is written as β1>0 ⇐⇒ max_i θ_i^4/c_i ≥ 1; because β_i=0 at equality, the correct threshold is strict inequality, consistent with Proposition 1.
  4. [Section 3.1, Proposition 2] Proposition 2 leaves the case β1>0, β2=0 to Appendix A.3; consider stating this edge case explicitly in the main text to avoid an apparent gap between the 'β2>0' and 'β1=0' clauses.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing results derive from external RMT results and explicit algebra; self-citations are contextual only.

full rationale

The derivation chain is self-contained relative to its stated inputs. The single-matrix eigenvector limits in Proposition 1 are quoted from external work [33] (Liu et al.), not from the authors' own prior results. Lemma 1's delocalization argument conditions on X2, making the residual direction w2(ω) a deterministic unit vector orthogonal to v for almost every ω; Proposition 1's second statement then applies pointwise, and the bounded indicator plus dominated convergence completes the proof. No uniformity over random orthogonal directions is imported from the authors' own work. The weighted Stack-SVD result (Theorem 1) is obtained by a change of variables that maps the optimization to the external heteroscedastic weighted-PCA problem of [29], and the optimal weights are derived analytically rather than fitted. The weighted SVD-Stack result (Theorem 2) is proved by solving a quadratically constrained program via the Sherman-Morrison formula, with attainability certified by an explicit weighting. The dominance claim (Theorem 3) compares two closed-form expressions using the monotonicity of f(x) and an elementary inequality; it does not reduce to a fitted input or to a self-citation. The self-citations that appear, e.g., [38], [4], [11], [36], and [37], are contextual references to applications, minimax background, or related methodology, and none of them is used as load-bearing support for the paper's asymptotic performance formulas, optimal weights, or dominance theorems. Consequently, no prediction is equivalent to its inputs by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard RMT theorems for spiked eigenvectors and on clearly stated model assumptions about a shared subspace and noise structure. No parameters are fitted to data; θ and c are model inputs, and the weights are derived analytically. No new entities are postulated.

assumptions (5)
  • domain assumption Entries of E_i are i.i.d. with mean 0, variance 1/d, and bounded fourth moment.
    Assumption 1 in Section 2, used to apply spiked eigenvector and delocalization theorems.
  • domain assumption Proportional asymptotics: n_i, d go to infinity with n_i/d approaching c_i in (0, infinity), and M is fixed.
    Assumption 2 in Section 2, defines the regime in which all asymptotic formulas are derived.
  • standard math Single-spike eigenvector asymptotics and delocalization from Ding (2020) and Liu et al. (2023).
    Used in Proposition 1 and Lemma 1 to characterize per-table singular vector overlaps and to show cross-table residual orthogonality.
  • standard math Optimally weighted PCA results of Hong et al. (2023), including their Lemma 6.2 and 6.3.
    Used in the proof of Theorem 1 to identify optimal Stack-SVD weights and the resulting performance equation.
  • domain assumption Rank-r model with shared subspace and, for the main corollaries, identical ordering of signal strengths across tables.
    Assumptions 3 and 4 in Section 6, used to extend rank-one results to multiple shared components.

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Cite this review

Pith. "Pith review of Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration." pith.science (2026). https://pith.science/paper/HLE4LOWY

@misc{pith2026250722170,
  author       = {Pith},
  title        = {Pith review of: Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLE4LOWY}},
  note         = {Machine review of arXiv:2507.22170}
}
read the original abstract

Modern data analysis increasingly requires identifying shared latent structure across multiple high-dimensional datasets. A commonly used model assumes that the data matrices are noisy observations of low-rank matrices with a shared singular subspace. In this case, two primary methods have emerged for estimating this shared structure, which vary in how they integrate information across datasets. The first approach, termed Stack-SVD, concatenates all the datasets, and then performs a singular value decomposition (SVD). The second approach, termed SVD-Stack, first performs an SVD separately for each dataset, then aggregates the top singular vectors across these datasets, and finally computes a consensus amongst them. While these methods are widely used, they have not been rigorously studied in the proportional asymptotic regime, which is of great practical relevance in today's world of increasing data size and dimensionality. This lack of theoretical understanding has led to uncertainty about which method to choose and limited the ability to fully exploit their potential. To address these challenges, we derive exact expressions for the asymptotic performance and phase transitions of these two methods and develop optimal weighting schemes to further improve both methods. Our analysis reveals that while neither method uniformly dominates the other in the unweighted case, optimally weighted Stack-SVD dominates optimally weighted SVD-Stack. We extend our analysis to accommodate multiple shared components, and provide practical algorithms for estimating optimal weights from data, offering theoretical guidance for method selection in practical data integration problems. Extensive numerical simulations and semi-synthetic experiments on genomic data corroborate our theoretical findings.

Figures

Figures reproduced from arXiv: 2507.22170 by the authors.

Figure 1
Figure 1. Schematic showing the methods Stack-SVD and SVD-Stack. is commonly modeled by assuming the matrices Xm are noisy observations of low-rank matrices with a shared right singular subspace V ∈ R d×r , with r ≪ d [2, 45, 15, 26, 56]. In the case of a single data matrix X ∈ R n×d , the truncated singular value decomposi￾tion (SVD) produces Uˆ ∈ R n×r , Θˆ ∈ R r×r and Vˆ ∈ R d×r , that minimizes the Frobenius norm reconstr… view at source ↗
Figure 2
Figure 2. For each mean µ ∈ {0, .4, .7}, 100 values of c 1/4 and θ were sampled according to the procedure specified in the main text. For each fixed value of c and θ, 10 replicates were generated and |⟨v, v ˆ ⟩|2 was compared to its theoretical prediction to obtain esti￾mates of the bias (a) and standard error (b). 7. Experiments. We provide extensive simulations on synthetic data, demon￾strating the close correspondence acr… view at source ↗
Figure 3
Figure 3. Empirical results converge to theoretical predictions for [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Simulations highlighting several illustrative examples. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: a. The top two left singular vectors obtained from the (transformed) count matrices for three different levels of (artificially) introduced ambient RNA. The color of each cell corresponds to the cell-type assigned by [57]. b. and c. The low and high noise scenarios wer…

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Reference graph

Works this paper leans on

61 extracted references · 54 canonical work pages · cited by 4 Pith papers

  1. [33]

    and Zhang, Z

    Liu, X., Liu, Y., Pan, G., Zhang, L. and Zhang, Z. (2023). Asymptotic properties of spiked eigenvalues and eigenvectors of signal-plus-noise matrices with their applications. arXiv preprint arXiv:2310.13939

    arXiv 2023

  2. [29]

    Hong, D., Yang, F., Fessler, J. A. and Balzano, L. (2023). Optimally weighted PCA for high-dimensional heteroscedastic data.SIAM Journal on Mathematics of Data Science5 222–250

    work page 2023

  3. [17]

    Ding, X. (2020). High dimensional deformed rectangular matrices with applications in matrix denoising. Bernoulli 26 387–417

    work page 2020

  4. [1]

    and Stegun, I

    Abramowitz, M. and Stegun, I. A. (1965).Handbook of mathematical functions: with formulas, graphs, and mathematical tables55. Courier Corporation

    work page 1965

  5. [2]

    , Arnol, D

    Argelaguet, R. , Arnol, D. , Bredikhin, D. , Deloro, Y. , Velten, B. , Marioni, J. C. and Stegle, O. (2020). MOFA+: a statistical framework for comprehensive integration of multi-modal single-cell data.Genome biology 21 1–17

    work page 2020

  6. [3]

    Arroyo, J., Athreya, A., Cape, J., Chen, G., Priebe, C. E. and Vogelstein, J. T. (2021). Inference for multiple heterogeneous networks with a common invariant subspace.Journal of Machine Learning Research22 1–49

    work page 2021

  7. [4]

    Z., Tse, D

    Baharav, T. Z., Tse, D. and Salzman, J. (2024). OASIS: An interpretable, finite-sample valid alternative to Pearson’s X 2 for scientific discovery.Proceedings of the National Academy of Sciences 121

    work page 2024

  8. [5]

    , Ben Arous, G

    Baik, J. , Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.The Annals of Probability33 1643–1697. https://doi.org/10.1214/009117905000000233

    doi:10.1214/009117905000000233 2005

Show all 61 references
  1. [6]

    Bartlett, M. S. (1947). The use of transformations.Biometrics 3 39–52

    1947

  2. [7]

    and Nadakuditi, R

    Benaych-Georges, F. and Nadakuditi, R. R. (2012). The singular values and vectors of low rank perturbations of large rectangular random matrices.Journal of Multivariate Analysis 111 120–135

    2012

  3. [8]

    and Wijewardena, P

    Bhaskara, A. and Wijewardena, P. M. (2019). On distributed averaging for stochastic k-pca. Advances in neural information processing systems32

    2019

  4. [9]

    Bunch, J. R. , Nielsen, C. P. and Sorensen, D. C. (1978). Rank-one modification of the symmetric eigenproblem.Numerische Mathematik 31 31–48

    1978

  5. [10]

    Cai, T. T. , Han, X. and Pan, G. (2020). Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

    2020

  6. [11]

    Chaung, K., Baharav, T. Z. , Henderson, G. , Zheludev, I. N. , W ang, P. L. and Salz- man, J. (2023). SPLASH: A statistical, reference-free genomic algorithm unifies biological discovery.Cell 186 5440–5456

    2023

  7. [12]

    , Ardlie, K

    Consortium, G. , Ardlie, K. G. , Deluca, D. S. , Segrè, A. V. , Sullivan, T. J. , Young, T. R. , Gelfand, E. T. , Trowbridge, C. A. , Maller, J. B. , Tukiainen, T. et al. (2015). The Genotype-Tissue Expression (GTEx) pilot analysis: multitissue gene reg- ulation in humans.Scie...

    2015

  8. [13]

    Consortium*, T. T. S. , Jones, R. C. , Karkanias, J. , Krasnow, M. A. , Pisco, A. O. , Quake, S. R. , Salzman, J. , Yosef, N. , Bulthaup, B., Brown, P. et al. (2022). The Tabula Sapiens: A multiple-organ, single-cell transcriptomic atlas of humans.Science 376 eabl4896

    2022

  9. [14]

    Danning, R., Hu, F. B. and Lin, X. (2025). LACE-UP: An ensemble machine-learning method for health subtype classification on multidimensional binary data.Proceedings of the Na- tional Academy of Sciences122

    2025

  10. [15]

    , Bellio, R., Trippa, L

    De Vito, R. , Bellio, R., Trippa, L. and Parmigiani, G. (2021). Bayesian multistudy factor analysis for high-throughput biological data.The annals of applied statistics15 1723–1741

    2021

  11. [16]

    Dence, T. P. and Dence, J. B. (2009). A survey of Euler’s constant.Mathematics Magazine 82 255–265

    2009

  12. [18]

    and Ma, R

    Ding, X. and Ma, R. (2025). Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators. Journal of the American Statistical As- sociations to appear

    2025

  13. [19]

    and Lopes, M

    Dörnemann, N. and Lopes, M. E. (2025).Tracy-Widom,Gaussian,andBootstrap:Approxima- tions for Leading Eigenvalues in High-Dimensional PCA.arXiv preprint arXiv:2503.23097

    2025 arXiv

  14. [20]

    Durrett, R. (2019). Probability: theory and examples49. Cambridge university press

    2019

  15. [21]

    and Young, G

    Eckart, C. and Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika 1 211–218

    1936

  16. [22]

    and Zhu, Z

    F an, J., W ang, D., W ang, K. and Zhu, Z. (2019). Distributed estimation of principal eigenspaces. Annals of statistics47 3009

    2019

  17. [23]

    Fleming, S. J. , Chaffin, M. D. , Arduini, A., Akkad, A.-D., Banks, E., Marioni, J. C. , Philippakis, A. A. , Ellinor, P. T. and Babadi, M. (2023). Unsupervised removal of systematic background noise from droplet-based single-cell experiments using CellBender. Nature methods20...

    2023

  18. [24]

    , Zhou, D

    Gan, Z. , Zhou, D. , Rush, E. , Panickan, V. A. , Ho, Y.-L. , Ostrouchovm, G. , Xu, Z. , Shen, S., Xiong, X., Greco, K. F. et al. (2025). Arch: Large-scale knowledge graph via aggregated narrative codified health records analysis. Journal of Biomedical Informatics 104761

    2025

  19. [25]

    Golub, G. H. and V an Loan, C. F. (2013). Matrix computations. JHU press

    2013

  20. [26]

    Grabski, I. N. , De Vito, R. , Trippa, L. and Parmigiani, G. (2023). Bayesian combinatorial MultiStudy factor analysis.The annals of applied statistics17 2212

    2023

  21. [27]

    and W ang, Y.(2024)

    He, Y., Liu, Z. and W ang, Y.(2024). Distributed Learning for Principal Eigenspaces without Moment Constraints.Journal of Computational and Graphical Statistics1–22

    2024

  22. [28]

    and Berger, B

    Hie, B., Bryson, B. and Berger, B. (2019). Efficient integration of heterogeneous single-cell transcriptomes using Scanorama.Nature biotechnology37 685–691

    2019

  23. [30]

    Johnstone, I. M. and Paul, D. (2018). PCA in high dimensions: An orientation.Proceedings of the IEEE106 1277–1292

    2018

  24. [31]

    and Eguchi, S

    Jou, Z.-Y., Huang, S.-Y., Hung, H. and Eguchi, S. (2024). A Generalized Mean Approach for Distributed-PCA.arXiv preprint arXiv:2410.00397

    2024 arXiv

  25. [32]

    , Kluger, Y

    Landa, B. , Kluger, Y. and Ma, R. (2024). Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets.arXiv preprint arXiv:2407.01718

    2024 arXiv

  26. [34]

    and Vivo, P

    Livan, G., Novaes, M. and Vivo, P. (2018). Introduction to random matrices theory and practice. Monograph Award63 54–57

    2018

  27. [35]

    Löffler, M., Zhang, A. Y. and Zhou, H. H. (2021). Optimality of spectral clustering in the Gaussian mixture model.The Annals of Statistics49 2506–2530

    2021

  28. [36]

    Ma, R., Sun, E. D. , Donoho, D. and Zou, J. (2024). Principled and interpretable alignability testing and integration of single-cell data.Proceedings of the National Academy of Sciences 121 e2313719121

    2024

  29. [37]

    Ma, R., Sun, E. D. and Zou, J. (2023). A spectral method for assessing and combining multiple data visualizations.Nature Communications14 780. 22

    2023

  30. [38]

    and Ma, R

    Ma, Z. and Ma, R. (2024). Optimal estimation of shared singular subspaces across multiple noisy matrices.arXiv preprint arXiv:2411.17054

    2024 arXiv

  31. [39]

    , Parekh, A

    Ozbay, S. , Parekh, A. and Singh, R. (2023). Navigating the manifold of single-cell gene coexpression to discover interpretable gene programs.bioRxiv 2023–11

    2023

  32. [40]

    Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica1617–1642

    2007

  33. [41]

    Regev, A., Teichmann, S. A. , Lander, E. S. , Amit, I., Benoist, C., Birney, E., Boden- miller, B., Campbell, P., Carninci, P., Clatworthy, M. et al. (2017). The human cell atlas. elife 6 e27041

    2017

  34. [42]

    J., Gorski, S

    Saliba, A.-E., Westermann, A. J., Gorski, S. A. and Vogel, J. (2014). Single-cell RNA-seq: advances and future challenges.Nucleic acids research42 8845–8860

    2014

  35. [43]

    and Stephens, M

    Sarkar, A. and Stephens, M. (2021). Separating measurement and expression models clarifies confusion in single-cell RNA sequencing analysis.Nature genetics 53 770–777

    2021

  36. [44]

    and Gaynanova, I

    Sergazinov, R., Taeb, A. and Gaynanova, I. (2024). A spectral method for multi-view sub- space learning using the product of projections.arXiv preprint arXiv:2410.19125

    2024 arXiv

  37. [45]

    and Mo, Q

    Shen, R., W ang, S. and Mo, Q. (2012). Sparse integrative clustering of multiple omics data sets. The annals of applied statistics7 269

    2012

  38. [46]

    and Kontar, R

    Shi, N. and Kontar, R. (2024). Personalized pca: Decoupling shared and unique features. Journal of machine learning research25 1–82

    2024

  39. [47]

    , Hoffman, P

    Stuart, T., Butler, A. , Hoffman, P. , Hafemeister, C. , Papalexi, E., Mauck, W. M. , Hao, Y., Stoeckius, M., Smibert, P. and Satija, R. (2019). Comprehensive integration of single-cell data.cell 177 1888–1902

    2019

  40. [48]

    Tao, T. (2012). Topics in random matrix theory132. American Mathematical Soc

    2012

  41. [49]

    V an Loan, C. F. (1976). Generalizing the singular value decomposition. SIAM Journal on numerical Analysis13 76–83

    1976

  42. [50]

    Weinstein, J. N. , Collisson, E. A. , Mills, G. B. , Shaw, K. R. , Ozenberger, B. A. , Ellrott, K., Shmulevich, I., Sander, C. and Stuart, J. M. (2013). The cancer genome atlas pan-cancer analysis project.Nature genetics 45 1113–1120

    2013

  43. [51]

    Wold, H. (1985). Partial least squares.Encyclopedia of Statistical Sciences6 581–591

    1985

  44. [52]

    and Ma, C

    Yang, Y. and Ma, C. (2025). Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices.arXiv preprint arXiv:2501.09336

    2025 arXiv

  45. [53]

    Young, M. D. and Behjati, S. (2020). SoupX removes ambient RNA contamination from droplet-based single-cell RNA sequencing data.Gigascience 9 giaa151

    2020

  46. [54]

    Yu, Y., W ang, T.and Samworth, R. J. (2015). A useful variant of the Davis–Kahan theorem for statisticians.Biometrika 102 315–323

    2015

  47. [55]

    , Mathew, D

    Zhang, Z. , Mathew, D. , Lim, T. L. , Mason, K. , Martinez, C. M. , Huang, S. , Wherry, E. J. , Susztak, K., Minn, A. J. , Ma, Z. et al. (2024). Recovery of biological signals lost in single-cell batch integration with CellANOVA.Nature Biotechnology1–17

    2024

  48. [56]

    , Sun, H

    Zhang, Z. , Sun, H. , Mariappan, R. , Chen, X. , Chen, X. , Jain, M. S. , Efremova, M., Teichmann, S. A. , Rajan, V. and Zhang, X. (2023). scMoMaT jointly performs single cell mosaic integration and multi-modal bio-marker detection.Nature Communications 14 384

    2023

  49. [57]

    Zheng, G. X. , Terry, J. M. , Belgrader, P., Ryvkin, P., Bent, Z. W. , Wilson, R., Zi- raldo, S. B. , Wheeler, T. D. , McDermott, G. P. , Zhu, J. et al. (2017). Massively parallel digital transcriptional profiling of single cells.Nature communications8 14049

    2017

  50. [58]

    and Tang, M

    Zheng, R. and Tang, M. (2022). Limit results for distributed estimation of invariant subspaces in multiple networks inference and PCA.arXiv preprint arXiv:2206.04306. STACKED SVD OR SVD STACKED? 23 APPENDIX A: PROOFS FORSVD-STACK We compute the asymptotic performance of SVD-St...

    2022

  51. [59]

    Thus, λ1−λ2≥β1β2

    The largest eigenvector ofAβ can be derived using the Bunch-Nielsen-Sorensen for- mula [9], but to show that the largest eigenvalue is larger than 1 it suffices to takex as a normalized indicator over the entries corresponding toβ1,β 2, in which case the Rayleigh quotient x⊤Aβ...

  52. [60]

    +β2 0 11⊤ vmax(Aβ) = 1√ M 1 λmax(Aβ) = 1 + (M− 1)β2 0 β⊤vmax(Aβ) = √ Mβ 0 Thus the power of unweightedSVD-Stack when allθi and ci are equal is: |⟨v, ˆvSVD-Stack⟩|2 p − → Mβ 2 0 1 + (M− 1)β2 0 = M ( θ4 0−c0 ) Mθ 4 0 +θ2 0− (M− 1)c0 when θ4 0>c 0, which is equivalent to the stat...

  53. [61]

    An eigenvectorξ1 corresponding toγ1 satisfies (S20) (˜u0˜u⊤ 0 + Σ)ξ1 =γ1ξ1

    In particular, the dimension of the eigenspace corresponding tow2 1 must be at leastn1− 1, implying that the algebraic multiplicity ofw2 1 is at leastn1− 1. An eigenvectorξ1 corresponding toγ1 satisfies (S20) (˜u0˜u⊤ 0 + Σ)ξ1 =γ1ξ1. Under condition (S15), γ1 > maxiw2 i and thu...

    2000

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.