Pith. sign in

REVIEW 3 major objections 5 minor 49 references

Low Rank Convex Clustering For Matrix-Valued Observations

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Low-rank convex clustering provably recovers true matrix clusters, the paper claims, by adding a nuclear-norm penalty to a convex objective.

desk verdict A solid, mostly self-contained extension of convex clustering to matrix data; the recovery theorems require cliques while the experiments use kNN graphs, so the guarantees do not formally cover the demonstrated setting. read the letter →

arxiv 2412.17328 v1 pith:TU4JNGP4 submitted 2024-12-23 math.OC math.STstat.TH

Meixia Lin , Yangjing Zhang This is my paper · ORCID
classification math.OCmath.STstat.TH MSC 90C9090C2562H30
keywords lowrankconvexclusteringmatrix-valuedobservationsexactclusterrecoveryasymptoticpredictionerrorboundnuclearnormregularizationaugmentedLagrangianmethodsemismoothNewton
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces low-rank convex clustering (lrCC), an optimization model that extends convex clustering from vectors to matrix-valued observations by adding a nuclear-norm penalty that encourages each centroid to be low rank. The central theoretical aim is to show that this convex model can recover true cluster memberships exactly for finite samples and asymptotically as the sample size grows, under explicit conditions on the graph, edge weights, and penalty parameters. It also derives a finite-sample bound on centroid estimation error and shows prediction consistency for general graphs and weights. The authors develop a double-loop augmented-Lagrangian and semismooth-Newton algorithm so the model can actually be solved on image and matrix data sets. If these guarantees hold, lrCC gives a principled convex alternative to nonconvex low-rank k-means for matrix data such as images and spatio-temporal observations.

What carries the argument

The load-bearing construction is a partition-dependent centroid problem (P-Γ), which re-expresses (1.1) at the level of candidate clusters. Proposition 2.4 gives a sufficient merging condition: if each block of the partition induces a clique in the observation graph and γ1 is large enough relative to within-block distances, the unique solution of (1.1) assigns all points in the block a common centroid. Proposition 2.5 adds a distinguishing condition involving γ1, γ2, and the rank of the centroid, guaranteeing that two blocks with different means receive different centroids. For the statistical bound, the proof performs a singular value decomposition of D=B⊗Id, separating the d(n−κ0)-dimensional range from the dκ0-dimensional null space, then uses sub-Gaussian concentration and the nuclear-norm prox properties (the Frobenius deviation of Proxγ‖·‖∗ is at most γ√d2) to control the noise terms.

What would settle it

Run lrCC on two low-rank Gaussian clusters with a fully connected graph, uniform weights, and any penalty pair satisfying γ1≥γ1,min and γ1wmax+γ2√d2<Δ; Theorem 2.1 predicts perfect cluster recovery, so an adjusted Rand index below 1 would refute the exact-recovery claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the unique solution of the lrCC problem (1.1) provably recovers cluster structure: Theorem 2.1 gives sufficient conditions—within-cluster subgraphs being cliques, a lower bound on the fusion penalty γ1, and an upper bound γ1wmax+γ2√d2<Δ on the separation between cluster means—under which observations in each true cluster share a centroid and distinct clusters receive distinct centroids. Theorem 2.3 extends this to a Gaussian mixture setting, showing with high probability that all points within a tσ-ball around a low-rank mean are assigned to one cluster and balls from different means are separated. Theorem 3.1 bounds the average squared centroid error by σ²(κ0/n+√(κ0 log(dn)/(dn²))) plus γ1- and γ2-dependent terms, and the bound drives prediction consistency when the graph's incidence spectrum satisfies condition (3.4). The paper also shows the model is a convex relaxation of the low-rank k-means method for matrix observations.

Load-bearing premise

Exact recovery is proven only when every true cluster's within-cluster graph is a complete graph (a clique), but the demonstrations use nearest-neighbor graphs whose within-cluster subgraphs need not be complete.

Editorial extensions

If this is right

  • Exact membership recovery is guaranteed when within-cluster graphs are cliques and (γ1,γ2) lie in the explicitly computed region γ1≥γ1,min and γ1wmax+γ2√d2<Δ.
  • For a mixture of K isotropic Gaussians with low-rank means, all observations within a fixed radius tσ of their true mean are clustered together with probability at least 1−exp(−2ε²n), and different such groups are separated under the stated conditions.
  • Centroid estimation error converges to zero as n,d grow under condition (3.4), which holds for fully connected graphs when n=o(d/log d) and for k-nearest-neighbor graphs when n=o(√(d/log d)).
  • The double-loop algorithm converges globally, with Q-superlinear dual convergence and R-superlinear primal convergence when the penalty sequence σk→∞, making the model solvable at practical cost.

Reading between the lines

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

  • Theorem 2.1's clique condition is likely not necessary: on k-nearest-neighbor graphs the within-cluster subgraphs are far from cliques, so a testable extension is to replace the clique by a bounded-conductance or small-cut condition in Proposition 2.4.
  • The perfect region in (γ1,γ2) space suggests a simple tuning heuristic: set γ1 just above γ1,min and choose γ2 below (Δ−γ1wmax)/√d2; the paper does not propose this as an automated rule, but it follows directly from the theorem's inequalities.
  • Because the consistency condition favors n growing slower than √(d/log d) on k-NN graphs, the method is best suited to high-dimensional regimes; whether other weightings or adaptive graphs can lift that restriction is left open by the paper.
Share X Bluesky LinkedIn Reddit HN

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 proposes a low-rank convex clustering (lrCC) model for matrix-valued observations, extending convex clustering by adding a nuclear-norm penalty that promotes low-rank centroids. The main theoretical contributions are: (i) Theorem 2.1 gives finite-sample exact cluster recovery under conditions that include a within-cluster clique requirement on the graph; (ii) Theorem 2.3 extends this to asymptotic recovery of a t-ball around each low-rank mean for Gaussian mixtures; (iii) Theorem 3.1 provides a finite-sample prediction error bound for the centroid estimator, leading to prediction consistency under a condition on the incidence matrix; and (iv) a double-loop ALM with a semismooth Newton inner solver is developed and analyzed. Numerical experiments on synthetic and real data compare lrCC with two low-rank Lloyd variants, showing favorable ARI and NMI scores.

Significance. If the claims hold, lrCC is an early convex low-rank matrix clustering method with explicit exact-recovery and error bounds, and the algorithmic contribution is a practically oriented solver with superlinear local convergence guarantees. The proofs of Propositions 2.4, 2.5, Theorem 3.1, and the convergence theorems are largely self-contained and constitute a solid technical core. The extension of the prediction error analysis to M-concentrated noise in Appendix C is a useful generality. However, the recovery guarantees are proven only for graphs whose within-cluster subgraphs are cliques, while the experiments operate on k-nearest-neighbor graphs; this scope gap, together with an error in Appendix A.1, requires attention before the results can be accepted at face value.

major comments (3)
  1. [Theorem 2.1(a), Theorem 2.3(a1), and Section 5] The exact and asymptotic recovery guarantees are conditional on the induced subgraph on each true cluster (or t-ball) being a clique. This is not a stylistic assumption: the proof of Proposition 2.4 in Appendix B.1 constructs within-cluster subgradients only for edges in E and then replaces the sum with a sum over all pairs in J_alpha, an identity that holds only when J_alpha is a clique. The experiments in Section 5 use a 50-nearest-neighbor graph; for n=5000 with 2500 points per cluster, the within-cluster subgraph is far from a clique, so the theoretical recovery guarantees do not formally cover the demonstrated setting. The paper should either extend the recovery analysis to sparse graphs under appropriate weight and connectivity conditions, or state clearly that the finite-sample and asymptotic recovery results apply to dense (clique) graphs and that the kNN experiments are heuristic support.
  2. [Appendix A.1 and discussion after Eq. (2.2)] The bound 'max_{i in [n]} ||A_i - M_{s*_i}||_F = O(sigma sqrt(K) polylog(d/pi_min))' is derived from ||A - C||_F = O(sigma sqrt(nK) polylog(d/pi_min)). Since ||A - C||_F bounds the per-column deviations only up to a factor sqrt(n), the correct bound is O(sigma sqrt(nK) polylog(d/pi_min)). The missing sqrt(n) factor understates the required separation in the heuristic discussion of condition (2.2). This does not affect the statements of Theorems 2.1 and 2.3, but it should be corrected because it is used to motivate the asymptotic recovery result.
  3. [Section 3.1, statement preceding Theorem 3.1] The assumption 'Without loss of generality, we assume min_{l(i,j) in E} w_ij >= 1/2' is not a genuine WLOG reduction. The theorem's lower bound on gamma_1 and the final bound are not invariant under rescaling of the weights: if all weights are multiplied by c and gamma_1 is divided by c, the objective is unchanged, but the threshold gamma_1 >= 4 sigma / sigma_min(B) sqrt(d log(d|E|)) changes by a factor of 1/c. Thus the theorem applies only to weight configurations with minimum weight at least 1/2. This restriction should be stated as an explicit assumption, or the bound should be reformulated in a scale-invariant manner.
minor comments (5)
  1. [Title and abstract] The title in the full text contains spacing errors: 'MA TRIX-V ALUED OBSER V A TIONS' should read 'MATRIX-VALUED OBSERVATIONS'.
  2. [Section 5.4, paragraph after Fig. 8] The sentence 'The ARI and NMI scores for each method across these tasks are displayed in Figure 8' should refer to Figure 9, since Figure 8 shows sample images, not the performance curves.
  3. [Proof of Theorem 2.3(ii), Section 2.3] The phrase 'The optimality condition of (1.1) is' is not followed by a displayed equation; the sentence is incomplete and should be revised.
  4. [Section 2.2, notation after Theorem 2.3] The notation 'e wmax' and 'ewmax' are used inconsistently within the same paragraph; one symbol should be chosen and used throughout.
  5. [Section 3.1, Remark 3.2] For the k-nearest-neighbor graph, the sufficient condition 'n = o(sqrt(d/log d))' is more conservative than what follows from the authors' own Lemma A.1; a direct computation yields sqrt(k n log d / d) = o(1), i.e., n = o(d/log d). The weaker condition is not an error, but the remark could be sharpened.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the recovery guarantees and error bounds are derived from the lrCC objective via subgradient optimality conditions and independent external results.

full rationale

The paper's main theoretical chain is self-contained: Theorem 2.1 follows from Propositions 2.4 and 2.5, whose proofs verify that the constructed subgradients satisfy the optimality conditions of (1.1) under stated graph and penalty conditions; Theorem 2.3(i) is proven by applying Proposition 2.4 to the random set I_alpha,t, and Theorem 2.3(ii) explicitly builds on the independent external result [15, Theorem 3] together with the paper's own optimality bound ||X*_i - A_i||_F <= gamma1(n-1)w_max + gamma2 sqrt(d2). Theorem 3.1 is proven by an explicit oracle-type inequality in Section 3.3, using SVD of the stacked difference operator, concentration inequalities, and the stated condition gamma1 >= 4 sigma sqrt(d log(d|E|))/sigma_min(B). No fitted parameter is relabeled as a prediction, and no uniqueness or modeling ansatz is imported from the authors' own prior work as a load-bearing premise. The only self-citations are [44,45] in the concluding future-work remark about adaptive sieving, which is not load-bearing for any theorem, experiment, or model justification. The gap between the clique requirement in condition (a) and the kNN graphs used in some experiments is a scope limitation of the conditional theorems, not a circular step: the theorems remain valid conditional statements, and the small synthetic validation in Section 2.1 uses a 50-sample-per-cluster setting where the induced subgraph is a clique. Thus the derivation chain does not reduce to its own inputs, and the paper is honest about its conditional guarantees.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim relies on a low-rank Gaussian mixture model, a user-supplied graph with clique-like within-cluster structure, two manually tuned penalty parameters, and standard convex-analysis and concentration results. No new physical or mathematical entities are postulated. The main contribution is the model and its analysis, not a new object.

free parameters (3)
  • gamma1 (fusion penalty weight) = Dataset-dependent: 8, 0.08, 10^-0.5, 1, 0.5, 0.0026 in experiments
    Controls how strongly linked centroids are fused; chosen by hand per experiment and fixed in theory only through inequalities.
  • gamma2 (nuclear norm penalty weight) = Dataset-dependent: 8/15, 0.04, 10^-1.5, 0.1, 0.05, 0.8 in experiments
    Controls low-rank shrinkage; set manually with no automatic selection procedure.
  • Graph parameters k and kernel bandwidth = k = 50 nearest neighbors; Gaussian kernel exp(-0.5 ||Ai - Aj||_F^2)
    The edge set and weights are user-chosen; all recovery guarantees depend on this graph.
assumptions (4)
  • domain assumption The true centroids of the clusters are low-rank matrices.
    The model (1.1) adds a nuclear norm penalty to promote low rank; Section 1 and Section 2.2 assume this for the low-rank mixture model.
  • domain assumption Observations are generated as Ai = M_{s_i*} + Ei with i.i.d. sub-Gaussian or Gaussian entries, the low-rank Gaussian mixture model.
    Used in Theorem 2.3 and the consistency argument in Section 3.2; the model is stated in (2.1).
  • domain assumption Within-cluster induced subgraphs of the graph G are cliques for exact recovery, and weights satisfy min w_ij >= 1/2 in Theorem 3.1.
    These restrict the applicability of the main recovery and error bounds; if violated, the stated guarantees do not hold.
  • standard math Standard convex analysis results: subdifferentials of nuclear and Frobenius norms, proximal mappings, Hanson-Wright and Hoeffding inequalities.
    Used throughout the proofs in Sections 2.3, 3.3, and Appendices B-D; no new axioms are introduced.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Low Rank Convex Clustering For Matrix-Valued Observations." pith.science (2026). https://pith.science/paper/TU4JNGP4

@misc{pith2026241217328,
  author       = {Pith},
  title        = {Pith review of: Low Rank Convex Clustering For Matrix-Valued Observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TU4JNGP4}},
  note         = {Machine review of arXiv:2412.17328}
}
abstract

Common clustering methods, such as $k$-means and convex clustering, group similar vector-valued observations into clusters. However, with the increasing prevalence of matrix-valued observations, which often exhibit low rank characteristics, there is a growing need for specialized clustering techniques for these data types. In this paper, we propose a low rank convex clustering model tailored for matrix-valued observations. Our approach extends the convex clustering model originally designed for vector-valued data to classify matrix-valued observations. Additionally, it serves as a convex relaxation of the low rank $k$-means method proposed by Z. Lyu, and D. Xia (arXiv:2207.04600). Theoretically, we establish exact cluster recovery for finite samples and asymptotic cluster recovery as the sample size approaches infinity. We also give a finite sample bound on prediction error in terms of centroid estimation, and further establish the prediction consistency. To make the model practically useful, we develop an efficient double-loop algorithm for solving it. Extensive numerical experiments are conducted to show the effectiveness of our proposed model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 48 canonical work pages

  1. [1]

    R. B. Bapat, Graphs and Matrices , Springer Science & Business Media, 2010

    work page 2010

  2. [2]

    Bizeul , On the log-Sobolev constant of log-concave measures , arXiv preprint arXiv:2306.12997, (2023)

    P. Bizeul , On the log-Sobolev constant of log-concave measures , arXiv preprint arXiv:2306.12997, (2023)

    work page arXiv 2023

  3. [3]

    J.-F. Cai, E. J. Cand `es, and Z. Shen , A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), pp. 1956–1982

    work page 2010

  4. [4]

    Chakraborty and J

    S. Chakraborty and J. Xu , Biconvex clustering , Journal of Computational and Graphical Statistics, 32 (2023), pp. 1524–1536. 36 MEIXIA LIN, AND YANGJING ZHANG

    work page 2023

  5. [5]

    E. C. Chi and K. Lange , Splitting methods for convex clustering , Journal of Computational and Graphical Statistics, 24 (2015), pp. 994–1013

    work page 2015

  6. [6]

    Y. Cui, D. F. Sun, and K.-C. Toh , On the R-superlinear convergence of the KKT residuals generated by the augmented Lagrangian method for convex composite conic programming , Mathematical Programming, 178 (2019), pp. 381–415

    work page 2019

  7. [7]

    Dasgupta, J

    A. Dasgupta, J. Hopcroft, R. Kannan, and P. Mitra , Spectral clustering with limited independence, in Proceedings of the eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 1036–1045

    work page 2007

  8. [8]

    N. M. M. De Abreu , Old and new results on algebraic connectivity of graphs , Linear Algebra and its Applications, 423 (2007), pp. 53–73

    work page 2007

Show all 49 references
  1. [9]

    Erd ˝os, J

    P. Erd ˝os, J. Pach, R. Pollack, and Z. Tuza , Radius, diameter, and minimum degree , Journal of Combinatorial Theory, Series B, 47 (1989), pp. 73–79

    1989

  2. [10]

    D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in indepen- dent random variables , The Annals of Mathematical Statistics, 42 (1971), pp. 1079–1083

    1971

  3. [11]

    G. E. Hinton and S. Roweis , Stochastic neighbor embedding, in Advances in Neural Informa- tion Processing Systems, vol. 15, 2002

    2002

  4. [12]

    T. D. Hocking, A. Joulin, F. Bach, and J.-P. Vert, Clusterpath an algorithm for clustering using convex fusion penalties , in International Conference on Machine Learning, 2011, pp. 745–752

    2011

  5. [13]

    Hubert and P

    L. Hubert and P. Arabie, Comparing partitions, Journal of Classification, 2 (1985), pp. 193– 218

    1985

  6. [14]

    Jiang, D

    K. Jiang, D. F. Sun, and K.-C. Toh , A partial proximal point algorithm for nuclear norm regularized matrix least squares problems , Mathematical Programming Computation, 6 (2014), pp. 281–325

    2014

  7. [15]

    Jiang, S

    T. Jiang, S. V avasis, and C. W. Zhai, Recovery of a mixture of Gaussians by sum-of-norms clustering, Journal of Machine Learning Research, 21 (2020), pp. 1–16

    2020

  8. [16]

    Kumar and R

    A. Kumar and R. Kannan, Clustering with spectral norm and the k-means algorithm , in 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 299–308

    2010

  9. [17]

    T. O. Kvalseth , Entropy and correlation: Some comments , IEEE Transactions on Systems, Man, and Cybernetics, 17 (1987), pp. 517–519

    1987

  10. [18]

    Ledoux , The Concentration of Measure Phenomenon , American Mathematical Society, 2001

    M. Ledoux , The Concentration of Measure Phenomenon , American Mathematical Society, 2001

    2001

  11. [19]

    X. Li, D. F. Sun, and K.-C. Toh , On efficiently solving the subproblems of a level-set method for fused lasso problems , SIAM Journal on Optimization, 28 (2018), pp. 1842–1866

    2018

  12. [20]

    Lindsten, H

    F. Lindsten, H. Ohlsson, and L. Ljung, Clustering using sum-of-norms regularization: With application to particle filter output computation, in 2011 IEEE Statistical Signal Processing Workshop (SSP), 2011, pp. 201–204

    2011

  13. [21]

    Lindsten, H

    F. Lindsten, H. Ohlsson, and L. Ljung , Just relax and come clustering!: A convexification of k-means clustering , tech. report, Link¨ oping University, 2011

    2011

  14. [22]

    Lloyd, Least squares quantization in PCM , IEEE Transactions on Information Theory, 28 (1982), pp

    S. Lloyd, Least squares quantization in PCM , IEEE Transactions on Information Theory, 28 (1982), pp. 129–137

    1982

  15. [23]

    F. J. Luque, Asymptotic convergence analysis of the proximal point algorithm , SIAM Journal on Control and Optimization, 22 (1984), pp. 277–293

    1984

  16. [24]

    Lyu and D

    Z. Lyu and D. Xia, Optimal clustering by Lloyd’s algorithm for low-rank mixture model , arXiv preprint arXiv:2207.04600, (2022)

    2022 arXiv

  17. [25]

    Lyu and D

    Z. Lyu and D. Xia , Optimal estimation and computational limit of low-rank Gaussian mix- tures, The Annals of Statistics, 51 (2023), pp. 646–667

    2023

  18. [26]

    S. Ma, D. Goldfarb, and L. Chen , Fixed point and Bregman iterative methods for matrix rank minimization, Mathematical Programming, 128 (2011), pp. 321–353

    2011

  19. [27]

    Merris, Multilinear Algebra, Crc Press, 1997

    R. Merris, Multilinear Algebra, Crc Press, 1997

    1997

  20. [28]

    Mohar, Y

    B. Mohar, Y. Alavi, G. Chartrand, and O. Oellermann , The Laplacian spectrum of graphs, Graph Theory, Combinatorics, and Applications, 2 (1991), pp. 871–898

    1991

  21. [29]

    Panahi, D

    A. Panahi, D. Dubhashi, F. D. Johansson, and C. Bhattacharyya , Clustering by sum of norms: Stochastic incremental algorithm, convergence and cluster recovery , in Interna- tional Conference on Machine Learning, 2017, pp. 2769–2777

    2017

  22. [30]

    Pelckmans, J

    K. Pelckmans, J. De Brabanter, J. A. Suykens, and B. De Moor , Convex clustering shrinkage, in PASCAL Workshop on Statistics and Optimization of Clustering Workshop, vol. 1524, 2005

    2005

  23. [31]

    Rezaei and P

    M. Rezaei and P. Fr ¨anti, Set matching measures for external cluster validity , IEEE Trans- actions on Knowledge and Data Engineering, 28 (2016), pp. 2173–2186

    2016

  24. [32]

    R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970

    1970

  25. [33]

    R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm LOW RANK CONVEX CLUSTERING 37 in convex programming, Mathematics of Operations Research, 1 (1976), pp. 97–116

    1976

  26. [34]

    Samson , Concentration of measure inequalities for Markov chains and ϕ-mixing pro- cesses, The Annals of Probability, 28 (2000), pp

    P.-M. Samson , Concentration of measure inequalities for Markov chains and ϕ-mixing pro- cesses, The Annals of Probability, 28 (2000), pp. 416–461

    2000

  27. [35]

    D. F. Sun, K.-C. Toh, and Y. Yuan , Convex clustering: Model, theoretical guarantee and efficient algorithm , Journal of Machine Learning Research, 22 (2021), pp. 1–32

    2021

  28. [36]

    K. M. Tan and D. Witten , Statistical properties of convex clustering , Electronic Journal of Statistics, 9 (2015), pp. 2324–2347

    2015

  29. [37]

    Tao and V

    T. Tao and V. Vu , Random matrices: The distribution of the smallest singular values , Geo- metric and Functional Analysis, 20 (2010), pp. 260–297

    2010

  30. [38]

    V an der Maaten and G

    L. V an der Maaten and G. Hinton, Visualizing data using t-SNE , Journal of Machine Learn- ing Research, 9 (2008), p. 2579–2605

    2008

  31. [39]

    Vu and K

    V. Vu and K. W ang , Random weighted projections, random quadratic forms and random eigenvectors, Random Structures & Algorithms, 47 (2015), pp. 792–821

    2015

  32. [40]

    W ang, Y

    B. W ang, Y. Zhang, W. W. Sun, and Y. F ang, Sparse convex clustering, Journal of Compu- tational and Graphical Statistics, 27 (2018), pp. 393–403

    2018

  33. [41]

    G. A. W atson, Characterization of the subdifferential of some matrix norms , Linear Algebra and its Applications, 170 (1992), pp. 33–45

    1992

  34. [42]

    Yang, A Study on Nonsymmetric Matrix-Valued Functions , Master Thesis, Department of Mathematics, National University of Singapore, (2009)

    Z. Yang, A Study on Nonsymmetric Matrix-Valued Functions , Master Thesis, Department of Mathematics, National University of Singapore, (2009)

    2009

  35. [43]

    Yuan and Y

    M. Yuan and Y. Lin , Model selection and estimation in regression with grouped variables , Journal of the Royal Statistical Society Series B: Statistical Methodology, 68 (2006), pp. 49– 67

    2006

  36. [44]

    Yuan, T.-H

    Y. Yuan, T.-H. Chang, D. F. Sun, and K.-C. Toh , A dimension reduction technique for large-scale structured sparse optimization problems with application to convex clustering , SIAM Journal on Optimization, 32 (2022), pp. 2294–2318

    2022

  37. [45]

    Y. Yuan, M. Lin, D. F. Sun, and K.-C. Toh , Adaptive sieving: A dimension reduction technique for sparse optimization problems , arXiv preprint arXiv:2306.17369, (2023)

    2023 arXiv

  38. [46]

    Zajkowski , Bounds on tail probabilities for quadratic forms in dependent sub-Gaussian random variables, Statistics & Probability Letters, 167 (2020), p

    K. Zajkowski , Bounds on tail probabilities for quadratic forms in dependent sub-Gaussian random variables, Statistics & Probability Letters, 167 (2020), p. 108898

    2020

  39. [47]

    Zhang, N

    Y. Zhang, N. Zhang, D. F. Sun, and K.-C. Toh , An efficient Hessian based algorithm for solving large-scale sparse group lasso problems , Mathematical Programming, 179 (2020), pp. 223–263

    2020

  40. [48]

    X.-Y. Zhao, D. F. Sun, and K.-C. Toh , A Newton-CG augmented Lagrangian method for semidefinite programming, SIAM Journal on Optimization, 20 (2010), pp. 1737–1765

    2010

  41. [49]

    C. Zhu, H. Xu, C. Leng, and S. Yan , Convex optimization procedure for clustering: Theo- retical revisit, in Advances in Neural Information Processing Systems, vol. 27, 2014

    2014

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.