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REVIEW 3 major objections 5 minor 25 references

Perfect Clustering in Very Sparse Diverse Multiplex Networks

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Pith's one-line read A tensor-based clustering algorithm provably recovers every layer label in very sparse diverse multiplex networks, at sparsity levels that match known computational lower bounds up to logarithms.

desk verdict A real tensor-based step toward sparse multiplex layer clustering, but Lemma 2 is false in the dictionary case, so the advertised all-different-loading result is not yet proven. read the letter →

arxiv 2507.19423 v1 pith:3KOJKHGJ submitted 2025-07-25 stat.ML cs.LGmath.STstat.MEstat.TH

Marianna Pensky This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.MEstat.TH MSC 62H3005C8015A69
keywords MultiplexNetworkHigherOrderOrthogonalIterationsClusteringDIMPLE-SGRDPGSignedGeneralizedRandomDotProductGraphperfecttensordecompositionsparsenetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tackles the problem of grouping the layers of a multiplex network when different groups of layers live in different latent subspaces and every layer has its own connection-probability matrix. Earlier layer-by-layer methods required each layer to be fairly dense, but the paper's claim is that pooling information across all layers with a tensor decomposition allows perfect clustering at much higher sparsity. The main result, Theorem 3, states that Algorithm 1 recovers the true layer labels with probability at least $1 - c n^{-\tau}$ once conditions involving $\rho_n n (n \wedge L)$ and $\rho_n n \sqrt L$ growing faster than powers of $\log n$ are met. These conditions are, up to logarithmic factors, the same as computational lower bounds proved for a simpler model, so the result suggests that layer clustering in this broad model is no longer a dense-network task.

What carries the argument

The load-bearing object is the mode-3 eigenvector matrix $W$ of the probability tensor, together with the relation $\widetilde{\mathcal P} = \Theta \times_1 U \times_2 U \times_3 W$ obtained by concatenating the group subspaces. $W$ has a block structure: rows corresponding to different groups are orthogonal, while rows within a group have inner products bounded away from zero when the variability assumptions hold. The algorithm's engine is a regularized Higher-Order Orthogonal Iteration, an iterative tensor decomposition that alternately updates estimates of the factor matrices, initialized by spectral estimates; this produces an estimator $\hat W$ close enough in the two-to-infinity norm that thresholding its Gram matrix and clustering the membership matrix recovers every label. The separation estimate $|\langle W(l_1,\cdot), W(l_2,\cdot)\rangle| \ge C M L^{-1}$ for same-group pairs is what makes the threshold $T(n,L) = M L^{-1} R(n,L)$ workable.

What would settle it

Construct a noiseless DIMPLE-SGRDPG tensor with two groups in which each layer's loading matrix is supported on a different, non-overlapping subset of coordinates while the collection still satisfies Assumption A5; compute the true Gram matrix entries $\langle W(l_1,\cdot), W(l_2,\cdot)\rangle$. If the minimal same-group value is not bounded below by $C M/L$, Lemma 2 fails and no threshold can separate the groups, showing that Assumptions A1-A6 alone do not guarantee perfect clustering.

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Extended reading notes

Core claim

The central claim is that the de-meaned probability tensor admits a low-rank Tucker structure $\widetilde{\mathcal P} = \Theta \times_1 U \times_2 U \times_3 W$, where $U$ spans the union of group subspaces and $W$ is a matrix of layer eigenvectors. Although the rows of $W$ are not constant within a group, Lemma 2 shows that their inner products are zero across groups and bounded below by $C M/L$ within a group, provided the loading matrices satisfy a variability condition together with one of two structural conditions. This turns layer clustering into a thresholding problem on the Gram matrix of an estimated $\hat W$: after obtaining $\hat W$ via regularized Higher-Order Orthogonal Iteration, Algorithm 1 keeps entries above a threshold and applies $k$-means to the leading singular vectors of the resulting membership matrix. Theorem 3 proves that this procedure is exactly correct with high probability, not merely consistent, under sparsity conditions that up to logarithms match the computational lower bound for a simpler multilayer stochastic block model.

Load-bearing premise

The entire proof depends on the premise that layers assigned to the same group share enough structure so that their rows in a certain eigenvector matrix are noticeably correlated; if their connection matrices vary without a common support or a balanced dictionary, that correlation can vanish even when all of the paper's formal assumptions hold.

Editorial extensions

If this is right

  • Layer clustering in this model no longer requires $\rho_n n / \log n \to \infty$; it succeeds when $\rho_n n (n \wedge L)$ and $\rho_n n \sqrt L$ dominate powers of $\log n$, so very sparse layers can be clustered as long as many layers or nodes are available.
  • The method exploits the number of layers: increasing $L$ improves precision, which layer-by-layer algorithms cannot do.
  • Because the Stochastic Block Model and the mixed multilayer stochastic block model are particular cases, the same guarantee covers those models and recovers, up to log factors, the thresholds of earlier results without requiring $L = O(n)$.
  • Once layer labels are recovered, downstream tasks such as subspace estimation and loading-matrix estimation can be handled by existing per-group techniques, so this closes the sparse part of the inference pipeline.

Reading between the lines

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

  • Editorial extension: the Gram-matrix mechanism is not tied to the signed Bernoulli noise model, so the argument should extend to any tensor whose mode-3 factor matrix is block-orthogonal with non-negligible within-block inner products and sub-Gaussian noise.
  • Editorial extension: the threshold $T(n,L)$ is set theoretically; a data-driven threshold based on the empirical distribution of off-diagonal entries of $\hat Y$ would be a natural testable extension, but the paper does not provide one.
  • Editorial extension: the comparison to a computational lower bound is made for a simpler model, so whether the logarithmic factors in this paper can be removed for the DIMPLE-SGRDPG model itself remains open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies clustering of layers in the DIverse MultiPLEx Signed Generalized Random Dot Product Graph (DIMPLE-SGRDPG) model, where all layers share the same node set, can be partitioned into groups with distinct ambient subspaces, and within each group the connectivity matrices may all differ. The proposed method, Algorithm 1, estimates the tensor factor matrix W via regularized HOOI (Algorithms 2 and 3), forms the Gram matrix Y of rows of the estimated W, thresholds its entries to obtain a membership matrix X, and applies k-means to the leading singular vectors of X. The main theoretical result, Theorem 3, claims perfect clustering with probability at least 1 - c n^{-tau} under Assumptions A1-A6, the extra conditions of Lemma 2, and sparsity conditions that the authors argue match, up to logarithmic factors, the computational lower bounds of Lei et al. (2024) for a simpler model. Simulations compare the new algorithm with a layer-per-layer baseline.

Significance. If the main result were established, it would be a substantial contribution: pooling information across layers to achieve perfect clustering in very sparse diverse multiplex networks, with rates that are optimal up to logarithmic factors for a related but simpler model. The paper contains a careful tensor decomposition, new two-to-infinity perturbation bounds via HOOI, and a nontrivial analysis of the signal strength of the core tensor. These technical developments are of independent interest. However, the advertised scope of the main theorem is not supported because a load-bearing lemma, Lemma 2, is internally inconsistent in the dictionary case, and Theorem 3 inherits this failure. The paper does not provide code or machine-checked proofs, but the proofs are detailed and the simulations are clearly described.

major comments (3)
  1. [Section 3.2, Lemma 2, Eq. (3.40)] Lemma 2 condition (b), the dictionary case, is internally inconsistent. When within a group there are t_m >= 2 dictionary types with balanced sizes, the rows of the matrix ~Phi^(m) take only t_m distinct values, so the left singular vectors of M3(G0^(m)) are the normalized type indicators: ~W^(m)(i,:) = e_t / sqrt(L_{t,m}) if row i belongs to type t. For two same-group layers of different types, <W(l1,:), W(l2,:)> = 0, which contradicts (3.40) that asserts |<W(l1,:), W(l2,:)>| >= C M / L > 0. The proof in Section 8.3 for the dictionary case only establishes the row-norm bound ||~W^(m)(i,:)|| >= C M^{1/2} L^{-1/2} and never establishes the pairwise inner-product lower bound. This is not a gap in the proof: the claimed bound is false in general under condition (b).
  2. [Section 5.2, Theorem 3, and Eq. (4.45)] Theorem 3 depends critically on Eq. (4.45), which is derived from Lemma 2. If (3.40) fails for pairs of different dictionary types inside the same group, then the matrix X = I(|Y| > T) in Algorithm 1 has zero blocks within a group. Consequently X is not the membership matrix for the M groups, and k-means on SVD_M(X) cannot be guaranteed to recover the groups. Thus the perfect clustering conclusion of Theorem 3 collapses for the dictionary case that the paper explicitly claims to cover in Example 2. The theorem is therefore not valid in its stated generality; the authors must either remove the dictionary case from the main claim or redesign the clustering step so that it does not rely on the false inner-product lower bound.
  3. [Section 3.3 and Section 8.4, Example 2] The justification of Example 2 in Section 8.4 verifies only that Assumption A5 holds, namely a lower bound on the singular values of Phi^(m), but does not verify the inner-product condition (3.40) that Lemma 2 requires. Example 2 is therefore not valid support for Theorem 3. The issue is not merely a missing detail: as shown above, the dictionary construction with multiple types produces orthogonal rows for different types, so no application of singular-value bounds can rescue (3.40). The salvageable part of Lemma 2 appears to be condition (a) (common support with rm equal to the size of the support), where the proof based on row norms and orthogonal V_F^(m) may be correct, but the paper needs to state Theorem 3 only under a condition that is actually sufficient.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: 'networ ks' should be 'networks'.
  2. [Section 1.2] In the text after Figure 1, 'the latter technique is as as g ood as pure guessing' contains repeated 'as' and misplaced spaces; please fix.
  3. [Section 1.3] The phrase 'we shall use standard tnsor identities' contains a typo: 'tnsor' should be 'tensor'.
  4. [Section 8.3, Proof of Lemma 2] In the printed proof, the matrix Phi^(m) is typeset as 'P hi(m)' in Eq. (A55); this makes the reading difficult and should be corrected.
  5. [Section 5.2, Theorem 3] The statement 'If assumptions of Lemma 2 hold' is vague because Lemma 2 has two alternative conditions, (a) and (b), with different conclusions. The theorem should specify which condition is being invoked and should state any additional restrictions needed for the proof.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: Theorem 3's conditions are assumptions, not fitted outputs; self-citations are auxiliary.

full rationale

The derivation of perfect clustering in Theorem 3 is conditional on Assumptions A1–A6, Lemma 2's separation bounds, condition (5.64), and R(n,L)->0. These are hypotheses about the generative model and target rates, not quantities fitted to the observed adjacency tensor. Algorithm 1's threshold T(n,L)=M L^{-1} R(n,L) is a theoretical rate, and the proof shows |Y_hat(l1,l2)-Y(l1,l2)| <= C M L^{-1} (log n)^{-1} R(n,L), which makes X=I(|Y_hat|>T) equal to the membership matrix once R(n,L)->0. This is a standard perturbation argument, not a tautology. The paper does invoke prior work by the same author (Pensky 2024a for the two-to-infinity perturbation theorem, Pensky 2024b for the model and ancillary lemmas), but those are used as stated technical ingredients with their own assumptions; the clustering conclusion does not reduce to them by construction. The comparison with Lei et al. (2024) is an external benchmark for sparsity thresholds, not an input to the proof. A separate correctness concern, which is not circularity, is that the dictionary case of Lemma 2 (condition (b)) appears to assert a within-group inner-product lower bound incompatible with the exact structure of W described in the proof; that is an internal mathematical gap rather than a fitted-input or self-citation circularity. Overall, no prediction in the paper is equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted to data; all constants in the bounds are absolute or depend only on tau. The main assumptions are A1-A6 plus the extra Lemma 2 structural conditions. The algorithms require M, K_m, and r as known inputs, and the threshold T depends on unspecified constants, which is a practical gap.

assumptions (8)
  • domain assumption A1: pi_m proportional to 1/M with c_pi/M <= pi_m <= c_bar_pi/M
    Guarantees group sizes are comparable; used in Lemma 5 for concentration of L_m.
  • domain assumption A2: rows of X(m) are i.i.d. from fm supported on the unit ball; max dimension K <= C_K min K_m
    Needed for concentration of sample covariances and for bounding incoherence of U.
  • domain assumption A3: covariance matrices satisfy c <= lambda_min(Sigma(m)) <= lambda_max(Sigma(m)) <= c_bar
    Ensures Sigma(m) is invertible so that U(m) can be related to the sample covariance; used throughout proofs.
  • domain assumption A4: B(l) = rho_n B0(l) with bounded singular values and rho_n^2 n^2 L >= C log n
    Defines the sparse regime and the signal lower bound used for initialization and HOOI contraction.
  • domain assumption A5: sigma_{r_m}^2(Phi(m)) >= C_Phi M^{-1} L for r_m <= K_m(K_m+1)/2
    Controls the singular values of the layer-loading matrix so that W rows separate across groups.
  • domain assumption Lemma 2 separation structure: either r_m equals the common support size of B0(l) in group m, or the B0(l) come from a fixed dictionary with balanced usage
    Yields the lower bound |<W(l1,:), W(l2,:)>| >= C M/L inside groups, needed for thresholding; not implied by A5.
  • domain assumption A6: (log n)^2 <= L <= n^{tau0}
    Ensures enough layers for pooling while keeping union bounds polynomial in n.
  • ad hoc to paper M, K_m, and r = rank(M3(G)) are known inputs to the algorithms
    Algorithms 1-3 take M, K, and r as inputs; no rank or group-number estimation is analyzed.

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Cite this review

Pith. "Pith review of Perfect Clustering in Very Sparse Diverse Multiplex Networks." pith.science (2026). https://pith.science/paper/3KOJKHGJ

@misc{pith2026250719423,
  author       = {Pith},
  title        = {Pith review of: Perfect Clustering in Very Sparse Diverse Multiplex Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KOJKHGJ}},
  note         = {Machine review of arXiv:2507.19423}
}
read the original abstract

The paper studies the DIverse MultiPLEx Signed Generalized Random Dot Product Graph (DIMPLE-SGRDPG) network model (Pensky (2024)), where all layers of the network have the same collection of nodes. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise matrices of connection probabilities can be all different. This setting includes majority of multilayer network models as its particular cases. The key task in this model is to recover the groups of layers with unique subspace structures, since the case where all layers of the network are embedded in the same subspace has been fairly well studied. Until now, clustering of layers in such networks was based on the layer-per-layer analysis, which required the multilayer network to be sufficiently dense. Nevertheless, in this paper we succeeded in pooling information in all layers together and providing a tensor-based methodology that ensures perfect clustering for a much sparser network. Our theoretical results, established under intuitive non-restrictive assumptions, assert that the new technique achieves perfect clustering under sparsity conditions that, up to logarithmic factors, coincide with the computational lower bound derived for a much simpler model.

Figures

Figures reproduced from arXiv: 2507.19423 by the authors.

Figure 1
Figure 1. The between-layer clustering errors of the tensor-based (solid lines) and the layer-per-layer analysis based technique (dash lines). Matrices X (m) are generated by the truncated normal distribution with σ = 1, µ = 0, M = 3 and Km = K = 3. The entries of B (l) are generated as uniform random numbers between c = −0.02 and d = 0.02. Left panel: varying n, L = 50 (red), L = 150 (black), L = 250 (blue), L = 350 (green).… view at source ↗
Figure 2
Figure 2. Tensor structure with M = 2, L = 6, L1 = 4, L2 = 2, K1 = K2 = K = 2, r1 = 3, r2 = 2, r = 5. Tensors are exhibited by stacking layers side by side. White color means zero value but other colors are used for objects’ identification only. Identical colors do not mean identical element values. Top left panel: tensors G (top row), Ge (1) (middle row) and Ge (2) (bottom row) (tensors’ layers stacked side by side). Top rig… view at source ↗
Figure 3
Figure 3. The between-layer clustering errors RBL in (5.51) of Algorithms 1 (solid lines) and 4 (dash lines). Matrices X (m) are generated by the truncated normal distribution with σ = 1, µ = 0, M = 3 and Km = K = 3. The entries of B (l) are generated as uniform random numbers between c = −0.02 and d = 0.02 (top), c = −0.03 and d = 0.03 (middle) and c = −0.05 and d = 0.05 (bottom). Left panels: varying n; L = 50 (red), L = 15… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The between-layer clustering errors RBL in (5.51) of Algorithms 1 (solid lines) and 4 (dash lines). Matrices X (m) are generated by the Dirichlet distribution with α = 0.1, M = 3 and Km = K = 3. The entries of B (l) are generated as uniform random numbers between c = −…
Figure 5
Figure 5. Figure 5: Tensor structure with M = 2, L = 6, L1 = 4, L2 = 2, K1 = K2 = K = 2, r1 = 3, r2 = 2, r = 5. In the left panels, white color means zero value but other colors are used for object identification only. In the right panels, different shades of color identify different valu…

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Works this paper leans on

25 extracted references · 8 canonical work pages

  1. [1]

    Agterberg and A

    J. Agterberg and A. Zhang. Estimating higher-order mixed memberships via the _ 2, tensor perturbation bound. ArXiv: 2212.08642, 2022

    arXiv 2022

  2. [2]

    Agterberg and A

    J. Agterberg and A. Zhang. Statistical inference for low-rank tensors: Heteroskedasticity, subgaussianity, and applications. ArXiv: 2410.06381, 2024

    arXiv 2024

  3. [3]

    Arroyo, A

    J. Arroyo, A. Athreya, J. Cape, G. Chen, C. E. Priebe, and J. T. Vogelstein. Inference for multiple heterogeneous networks with a common invariant subspace. Journal of Machine Learning Research, 22 0 (142): 0 1--49, 2021. URL http://jmlr.org/papers/v22/19-558.html

    work page 2021

  4. [4]

    Athreya, D

    A. Athreya, D. E. Fishkind, M. Tang, C. E. Priebe, Y. Park, J. T. Vogelstein, K. Levin, V. Lyzinski, Y. Qin, and D. L. Sussman. Statistical inference on random dot product graphs: a survey. Journal of Machine Learning Research, 18 0 (226): 0 1--92, 2018. URL http://jmlr.org/papers/v18/17-448.html

    work page 2018

  5. [5]

    T. T. Cai and A. Zhang. Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics . The Annals of Statistics, 46 0 (1): 0 60 -- 89, 2018. doi:10.1214/17-AOS1541. URL https://doi.org/10.1214/17-AOS1541

    doi:10.1214/17-aos1541 2018

  6. [6]

    X. Fan, M. Pensky, F. Yu, and T. Zhang. Alma: Alternating minimization algorithm for clustering mixture multilayer network. Journal of Machine Learning Research, 23 0 (330): 0 1--46, 2022. URL http://jmlr.org/papers/v23/21-0182.html

    work page 2022

  7. [7]

    Gupta and D

    A. Gupta and D. Nagar. Matrix Variate Distributions . Chapman and Hall/CRC, 1999. doi:https://doi.org/10.1201/9780203749289

    doi:10.1201/9780203749289 1999

  8. [8]

    B.-Y. Jing, T. Li, Z. Lyu, and D. Xia. Community detection on mixture multilayer networks via regularized tensor decomposition . The Annals of Statistics, 49 0 (6): 0 3181 -- 3205, 2021. doi:10.1214/21-AOS2079. URL https://doi.org/10.1214/21-AOS2079

    work page doi:10.1214/21-aos2079 2021

Show all 25 references
  1. [9]

    Jones and P

    A. Jones and P. Rubin-Delanchy. The multilayer random dot product graph. ArXiv: 2007.10455, 2020. doi:10.48550/ARXIV.2007.10455

    2007 doi arXiv

  2. [10]

    Z. T. Ke, F. Shi, and D. Xia. Community detection for hypergraph networks via regularized tensor power iteration. ArXiv:1909.06503, 2020

    1909 arXiv

  3. [11]

    T. G. Kolda and B. W. Bader. Tensor decompositions and applications. SIAM REVIEW, 51 0 (3): 0 455--500, 2009

    2009

  4. [12]

    Lei and K

    J. Lei and K. Z. Lin. Bias-adjusted spectral clustering in multi-layer stochastic block models. Journal of the American Statistical Association, 118 0 (544): 0 2433--2445, 2023. doi:10.1080/01621459.2022.2054817

    2023

  5. [13]

    J. Lei, A. R. Zhang, and Z. Zhu. Computational and statistical thresholds in multi-layer stochastic block models . The Annals of Statistics, 52 0 (5): 0 2431 -- 2455, 2024. doi:10.1214/24-AOS2441. URL https://doi.org/10.1214/24-AOS2441

    2024 doi

  6. [14]

    Y. Luo, G. Raskutti, M. Yuan, and A. R. Zhang. A sharp blockwise tensor perturbation bound for orthogonal iteration. Journal of Machine Learning Research, 22 0 (179): 0 1--48, 2021. URL http://jmlr.org/papers/v22/20-919.html

    2021

  7. [15]

    Z. Lyu, T. Li, and D. Xia. Optimal clustering of discrete mixtures: Binomial, poisson, block models, and multi-layer networks. ArXiv:2311.15598, 2023 a

    2023 arXiv

  8. [16]

    Z. Lyu, D. Xia, and Y. Zhang. Latent space model for higher-order networks and generalized tensor decomposition. Journal of Computational and Graphical Statistics, 32 0 (4): 0 1320--1336, 2023 b

    2023

  9. [17]

    M. Pensky. Davis-kahan theorem in the two-to-infinity norm and its application to perfect clustering. ArXiv:2411.11728, 2024 a

    2024 arXiv

  10. [18]

    M. Pensky. Signed diverse multiplex networks: Clustering and inference. ArXiv:2204.12087, 2024 b

    2024 arXiv

  11. [19]

    Pensky and Y

    M. Pensky and Y. Wang. Clustering of diverse multiplex networks. IEEE Transactions on Network Science and Engineering, 11 0 (4): 0 3441--3454, 2024. doi:10.1109/TNSE.2024.3374102

    2024

  12. [20]

    Rao and M

    C. Rao and M. Rao. Matrix Algebra and its Applications to Statistics and Econometrics. World Scientific Publishing Co., 1st edition, 1998

    1998

  13. [21]

    Rubin-Delanchy, J

    P. Rubin-Delanchy, J. Cape, M. Tang, and C. E. Priebe. A Statistical Interpretation of Spectral Embedding: The Generalised Random Dot Product Graph . Journal of the Royal Statistical Society Series B: Statistical Methodology, 84 0 (4): 0 1446--1473, 06 2022. ISSN 1369-7412. do...

    2022 doi

  14. [22]

    M. J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019. doi:10.1017/9781108627771

    2019 doi

  15. [23]

    D. Xia, A. R. Zhang, and Y. Zhou. Inference for low-rank tensors—no need to debias . The Annals of Statistics, 50 0 (2): 0 1220 -- 1245, 2022. doi:10.1214/21-AOS2146

    2022 doi

  16. [24]

    Zhang and D

    A. Zhang and D. Xia. Tensor svd: Statistical and computational limits. IEEE Transactions on Information Theory, 64 0 (11): 0 7311--7338, 2018. doi:10.1109/TIT.2018.2841377

    2018

  17. [25]

    Zheng and M

    R. Zheng and M. Tang. Limit results for distributed estimation of invariant subspaces in multiple networks inference and pca. ArXiv: 2206.04306, 2022. doi:10.48550/ARXIV.2206.04306

    2022 doi

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