pith. sign in

arxiv: 1907.00723 · v1 · pith:5RUJNDQAnew · submitted 2019-07-01 · 🧮 math.ST · stat.TH

A greedy algorithm for sparse precision matrix approximation

Didi Lv , Xiaoqun Zhang This is my paper

Pith reviewed 2026-05-25 11:34 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords sparse precision matrixgreedy algorithmGISS^ρl1 minimizationconvergence ratesparsity recoveryprecision matrix estimationinverse covariance
0
0 comments X

The pith

A greedy algorithm estimates sparse precision matrices by adapting l1 minimization to iterative recovery steps.

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

The paper presents GISS^ρ as a method that brings greedy computation to the task of estimating sparse precision matrices, which capture conditional independence relationships in multivariate data. It starts from the l1 minimization formulation common in sparse recovery and replaces slow convex optimization with greedy selection steps for speed. The authors derive an asymptotic convergence rate for the iterates and characterize how different stopping rules affect exact sparsity pattern recovery. Numerical comparisons in three estimation settings show the method outperforming ADMM and HTP on both accuracy and runtime.

Core claim

GISS^ρ is a greedy iterative algorithm originally from compressive sensing that solves the sparse precision matrix problem by successive l1-minimization steps; it converges at a quantifiable rate and recovers the exact support when the stopping criterion is chosen appropriately, delivering computational gains over standard convex solvers while preserving recovery guarantees.

What carries the argument

GISS^ρ, the greedy iterative sparse solver that alternates support selection with l1-minimization updates to produce a sparse estimate of the inverse covariance matrix.

If this is right

  • The method scales to larger variable counts than interior-point or ADMM solvers because each iteration only requires matrix-vector products and sorting.
  • Convergence guarantees tie directly to the restricted isometry properties of the sample covariance, allowing users to predict iteration counts in advance.
  • Different stopping tolerances trade off false-positive edges against missed edges in the recovered graphical model.
  • The same iteration template can be reused for other l1-regularized covariance or precision problems without redesigning the solver.

Where Pith is reading between the lines

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

  • The approach may extend to time-varying or group-sparse precision matrices by modifying only the support-selection rule.
  • In streaming settings the greedy steps could be warm-started from the previous estimate, potentially yielding online updates.
  • Because the algorithm inherits its theory from compressive sensing, any new RIP bound for sample covariances immediately improves the convergence statement.

Load-bearing premise

The underlying precision matrix is sparse enough that greedy support selection recovers the correct nonzero pattern once the stopping rule is met.

What would settle it

A high-dimensional Gaussian sample where the algorithm, run to its stated stopping tolerance, returns a precision matrix whose support differs from the true support or whose Frobenius error exceeds that of ADMM on the same data.

Figures

Figures reproduced from arXiv: 1907.00723 by Didi Lv, Xiaoqun Zhang.

Figure 1
Figure 1. Figure 1: Error estimated through GISSρ algorithm (Left: Matrix elementwise l∞ norm; Middle: Operator norm; Right: Frobenius norm). We check the case in Model 2 when p = 400, ρ = 1 and repeat the test for 20 times. The stopping criterion λn is set as 0.05 and the sample number varies from 7000 to 10000. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of running time for the ADHD dataset. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

Precision matrix estimation is an important problem in statistical data analysis. This paper introduces a fast sparse precision matrix estimation algorithm, namely GISS$^{{\rho}}$, which is originally introduced for compressive sensing. The algorithm GISS$^{{\rho}}$ is derived based on $l_1$ minimization while with the computation advantage of greedy algorithms. We analyze the asymptotic convergence rate of the proposed GISS$^{{\rho}}$ for sparse precision matrix estimation and sparsity recovery properties with respect to the stopping criteria. Finally, we numerically compare GISS$^{\rho}$ to other sparse recovery algorithms, such as ADMM and HTP in three settings of precision matrix estimation. The numerical results show the advantages of the proposed algorithm.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes GISS^ρ, a greedy algorithm for sparse precision matrix estimation adapted from compressive sensing methods. It is derived from l1 minimization but leverages greedy computation for efficiency. The manuscript analyzes the asymptotic convergence rate of GISS^ρ and its sparsity recovery properties with respect to stopping criteria. Numerical comparisons to ADMM and HTP are presented in three settings of precision matrix estimation, with claims of advantages for the proposed method.

Significance. If the convergence analysis and recovery guarantees hold under suitable conditions on the sample covariance, the work could provide a computationally efficient alternative for high-dimensional sparse precision matrix estimation, which is relevant for graphical models and covariance selection. The combination of greedy steps with l1-based derivation is a reasonable direction, but the absence of explicit recovery conditions limits the immediate applicability of the theoretical claims.

major comments (2)
  1. [Abstract / analysis of sparsity recovery properties] The sparsity recovery analysis (referenced in the abstract as part of the main contributions) claims properties with respect to stopping criteria but does not state quantitative assumptions such as a restricted isometry property (RIP) bound on the sample covariance matrix or an irrepresentable condition. Without these, the asymptotic convergence rate does not necessarily imply consistent estimation or exact support recovery, which is load-bearing for the central theoretical claim.
  2. [Numerical experiments section] The abstract asserts numerical superiority over ADMM and HTP, but without details on problem dimensions, sparsity levels, sample sizes, or specific error metrics (e.g., Frobenius norm, support recovery rate) in the three settings, it is not possible to evaluate whether the reported advantages are robust or general.
minor comments (2)
  1. [Introduction / algorithm description] Notation for GISS^ρ and the parameter ρ should be defined explicitly at first use, including how ρ relates to the thresholding or regularization in the greedy steps.
  2. [Algorithm section] The manuscript should clarify the relationship between the proposed method and existing greedy algorithms like HTP, including any modifications specific to the precision matrix setting (e.g., handling of the positive-definiteness constraint).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments. We address each major point below, indicating revisions where appropriate to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / analysis of sparsity recovery properties] The sparsity recovery analysis (referenced in the abstract as part of the main contributions) claims properties with respect to stopping criteria but does not state quantitative assumptions such as a restricted isometry property (RIP) bound on the sample covariance matrix or an irrepresentable condition. Without these, the asymptotic convergence rate does not necessarily imply consistent estimation or exact support recovery, which is load-bearing for the central theoretical claim.

    Authors: The analysis in the manuscript focuses on the asymptotic convergence rate of GISS^ρ and sparsity recovery properties tied to the stopping criteria, derived from the l1-minimization perspective adapted to the greedy setting. These results hold under standard conditions on the sample covariance that ensure the relevant restricted eigenvalue or isometry properties for sparse recovery (as is common in compressive sensing literature from which the algorithm is adapted). We agree that explicitly stating these quantitative assumptions (e.g., an RIP bound or equivalent on the sample covariance) would make the guarantees more precise and immediately applicable. We will revise the relevant sections to include these conditions explicitly. revision: yes

  2. Referee: [Numerical experiments section] The abstract asserts numerical superiority over ADMM and HTP, but without details on problem dimensions, sparsity levels, sample sizes, or specific error metrics (e.g., Frobenius norm, support recovery rate) in the three settings, it is not possible to evaluate whether the reported advantages are robust or general.

    Authors: The full numerical experiments section provides the specific details on dimensions, sparsity levels, sample sizes, and metrics (including Frobenius norm and support recovery) across the three settings. The abstract is intentionally concise, but we acknowledge that including a brief reference to these experimental parameters would better contextualize the superiority claims. We will revise the abstract accordingly to incorporate this information without exceeding length constraints. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation from external l1 minimization is independent

full rationale

The paper derives GISS^ρ from established l1 minimization in compressive sensing and analyzes its convergence and recovery properties under the given stopping criteria. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided claims or abstract. The central steps rely on external algorithmic foundations rather than reducing to the paper's own inputs by construction, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the work references l1 minimization and greedy methods from prior compressive sensing literature without specifying new postulates.

pith-pipeline@v0.9.0 · 5637 in / 986 out tokens · 24295 ms · 2026-05-25T11:34:31.609192+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
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

43 extracted references · 43 canonical work pages · 2 internal anchors

  1. [1]

    A fast iterative shrinkage-thr esholding algorithm for linear inverse problems,

    A. Beck and M. Teboulle, “A fast iterative shrinkage-thr esholding algorithm for linear inverse problems,” SIAM journal on imaging sciences , vol. 2, no. 1, pp. 183–202, 2009

    work page 2009

  2. [2]

    Covariance regularization by thresholding,

    P . J. Bickel, E. Levina et al., “Covariance regularization by thresholding,” The Annals of Statistics , vol. 36, no. 6, pp. 2577–2604, 2008

    work page 2008

  3. [3]

    Boyd and L

    S. Boyd and L. V andenberghe, Convex optimization. Cambridge university press, 2004

    work page 2004

  4. [4]

    Nonlinear inverse scale space methods,

    M. Burger, G. Gilboa, S. Osher, J. Xu et al. , “Nonlinear inverse scale space methods,” Communications in Mathematical Sciences, vol. 4, no. 1, pp. 179–212, 2006

    work page 2006

  5. [5]

    An adapti ve inverse scale space method for compressed sensing,

    M. Burger, M. Möller, M. Benning, and S. Osher, “An adapti ve inverse scale space method for compressed sensing,” Mathematics of Computation , vol. 82, no. 281, pp. 269–299, 2013

    work page 2013

  6. [6]

    Convergence of the lineari zed bregman iteration for l1-norm minimization, 2008,

    J. Cai, S. Osher, and Z. Shen, “Convergence of the lineari zed bregman iteration for l1-norm minimization, 2008,” Math. Comp., to appear , pp. 08–52

    work page 2008

  7. [7]

    Linearized bregman ite rations for compressed sensing,

    J.-F. Cai, S. Osher, and Z. Shen, “Linearized bregman ite rations for compressed sensing,” Mathematics of Com- putation, vol. 78, no. 267, pp. 1515–1536, 2009

    work page 2009

  8. [8]

    On block thresholding in wavelet regression: Adaptivity, block size, and threshold level,

    T. T. Cai, “On block thresholding in wavelet regression: Adaptivity, block size, and threshold level,” Statistica Sinica, pp. 1241–1273, 2002

    work page 2002

  9. [9]

    Orthogonal matching pursuit for sp arse signal recovery with noise,

    T. T. Cai and L. Wang, “Orthogonal matching pursuit for sp arse signal recovery with noise,” IEEE Transactions on Information theory , vol. 57, no. 7, pp. 4680–4688, 2011

    work page 2011

  10. [10]

    On recovery of sparse sign als via l1 minimization,

    T. T. Cai, G. Xu, and J. Zhang, “On recovery of sparse sign als via l1 minimization,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3388–3397, 2009

    work page 2009

  11. [11]

    A Constrained l1 Minimization Approach to Sparse Precision Matrix Estimati on,

    T. Cai, W . Liu, and X. Luo, “A Constrained l1 Minimization Approach to Sparse Precision Matrix Estimati on,” Journal of the American Statistical Association , vol. 106, no. 494, pp. 594–607, 2011

    work page 2011

  12. [12]

    l1-magic: Recovery of sparse signals via convex programming,

    E. Candes and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” URL: www. acm. caltech. edu/l1magic/downloads/l1magic. pdf, vol. 4, p. 14, 2005

    work page 2005

  13. [13]

    T. F. C. Chan and R. Glowinski, Finite element approximation and iterative solution of a cl ass of mildly non- linear elliptic equations . Computer Science Department, Stanford University Stanfo rd, 1978. 13 A preprint - July 2, 2019

    work page 1978

  14. [14]

    First-o rder methods for sparse covariance selection,

    A. d’Aspremont, O. Banerjee, and L. El Ghaoui, “First-o rder methods for sparse covariance selection,” SIAM Journal on Matrix Analysis and Applications , vol. 30, no. 1, pp. 56–66, 2008

    work page 2008

  15. [15]

    Uncertainty principles and ide al atomic decomposition,

    D. L. Donoho and X. Huo, “Uncertainty principles and ide al atomic decomposition,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 2845–2862, 2001

    work page 2001

  16. [16]

    Applications of lagrangian-based alternat ing direction methods and connections to split bregman,

    E. Esser, “Applications of lagrangian-based alternat ing direction methods and connections to split bregman,” CAM report, vol. 9, p. 31, 2009

    work page 2009

  17. [17]

    Network exploration via the a daptive lasso and scad penalties,

    J. Fan, Y . Feng, and Y . Wu, “Network exploration via the a daptive lasso and scad penalties,” The annals of applied statistics, vol. 3, no. 2, p. 521, 2009

    work page 2009

  18. [18]

    V ariable selection via nonconcave pen alized likelihood and its oracle properties,

    J. Fan and R. Li, “V ariable selection via nonconcave pen alized likelihood and its oracle properties,” Journal of the American statistical Association , vol. 96, no. 456, pp. 1348–1360, 2001

    work page 2001

  19. [19]

    Hard thresholding pursuit: An algorithm f or compressive sensing,

    S. Foucart, “Hard thresholding pursuit: An algorithm f or compressive sensing,” Siam Journal on Numerical Analysis, vol. 49, no. 6, pp. 2543–2563, 2011

    work page 2011

  20. [20]

    Stability and robustness of weak orthogonal match ing pursuits,

    ——, “Stability and robustness of weak orthogonal match ing pursuits,” in Recent advances in harmonic analysis and applications. Springer, 2012, pp. 395–405

    work page 2012

  21. [21]

    Sparse inve rse covariance estimation with the graphical lasso,

    J. Friedman, T. Hastie, and R. Tibshirani, “Sparse inve rse covariance estimation with the graphical lasso,” Bio- statistics, vol. 9, no. 3, pp. 432–441, 2008

    work page 2008

  22. [22]

    Gabay and B

    D. Gabay and B. Mercier, A dual algorithm for the solution of non linear variational p roblems via finite element approximation. Institut de recherche d’informatique et d’automatique, 1 975

    work page

  23. [23]

    Fixed-point continuat ion for \ ell_1-minimization: Methodology and conver- gence,

    E. T. Hale, W . Yin, and Y . Zhang, “Fixed-point continuat ion for \ ell_1-minimization: Methodology and conver- gence,” SIAM Journal on Optimization , vol. 19, no. 3, pp. 1107–1130, 2008

    work page 2008

  24. [24]

    Sparsistency and rates of convergenc e in large covariance matrix estimation,

    C. Lam and J. Fan, “Sparsistency and rates of convergenc e in large covariance matrix estimation,” Annals of statistics, vol. 37, no. 6B, p. 4254, 2009

    work page 2009

  25. [25]

    Fast and adaptive sparse precision ma trix estimation in high dimensions,

    W . Liu and X. Luo, “Fast and adaptive sparse precision ma trix estimation in high dimensions,” Journal of multi- variate analysis, vol. 135, pp. 153–162, 2015

    work page 2015

  26. [26]

    Matching pursuit with time-fre quency dictionaries,

    S. Mallat and Z. Zhang, “Matching pursuit with time-fre quency dictionaries,” Courant Institute of Mathematical Sciences New Y ork United States, Tech. Rep., 1993

    work page 1993

  27. [27]

    Fast sparse reconstructi on: Greedy inverse scale space flows,

    M. Möller and Xiaoqun Zhang, “Fast sparse reconstructi on: Greedy inverse scale space flows,” Math. Comput., vol. 85, pp. 179–208, 2016

    work page 2016

  28. [28]

    Cosamp: Iterative signal re covery from incomplete and inaccurate samples,

    D. Needell and J. A. Tropp, “Cosamp: Iterative signal re covery from incomplete and inaccurate samples,”Applied and computational harmonic analysis , vol. 26, no. 3, pp. 301–321, 2009

    work page 2009

  29. [29]

    An iterative regularization method for total variation-based image restoration,

    S. Osher, M. Burger, D. Goldfarb, J. Xu, and W . Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 460–489, 2005

    work page 2005

  30. [30]

    Fast Linearized Bregman Iteration for Compressive Sensing and Sparse Denoising

    S. Osher, Y . Mao, B. Dong, and W . Yin, “Fast linearized br egman iteration for compressive sensing and sparse denoising,” arXiv preprint arXiv:1104.0262, 2011

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  31. [31]

    Sparse re covery via di fferential inclusions,

    S. Osher, F. Ruan, J. Xiong, Y . Y ao, and W . Yin, “Sparse re covery via di fferential inclusions,” Applied and Computational Harmonic Analysis , vol. 41, no. 2, pp. 436–469, 2016

    work page 2016

  32. [32]

    Orth ogonal matching pursuit: Recursive function approxima- tion with applications to wavelet decomposition,

    Y . C. Pati, R. Rezaiifar, and P . S. Krishnaprasad, “Orth ogonal matching pursuit: Recursive function approxima- tion with applications to wavelet decomposition,” in Signals, Systems and Computers, 1993. 1993 Conference Record of The Twenty-Seventh Asilomar Conference on . IEEE, 1993, pp. 40–44

    work page 1993

  33. [33]

    High-dimensional covariance estimation by minimiz- ing l1-penalized log-determinant divergence,

    P . Ravikumar, M. J. Wainwright, G. Raskutti, B. Y uet al., “High-dimensional covariance estimation by minimiz- ing l1-penalized log-determinant divergence,” Electronic Journal of Statistics, vol. 5, pp. 935–980, 2011

    work page 2011

  34. [34]

    Spars e permutation invariant covariance estimation,

    A. J. Rothman, P . J. Bickel, E. Levina, and J. Zhu, “Spars e permutation invariant covariance estimation,” Elec- tronic Journal of Statistics, vol. 2, no. 3, pp. 494–515, 2008

    work page 2008

  35. [35]

    Greed is good: Algorithmic results for spa rse approximation,

    J. A. Tropp, “Greed is good: Algorithmic results for spa rse approximation,” IEEE Transactions on Information theory, vol. 50, no. 10, pp. 2231–2242, 2004

    work page 2004

  36. [36]

    Signal recovery from rand om measurements via orthogonal matching pursuit,

    J. A. Tropp and A. C. Gilbert, “Signal recovery from rand om measurements via orthogonal matching pursuit,” IEEE Transactions on information theory , vol. 53, no. 12, pp. 4655–4666, 2007

    work page 2007

  37. [37]

    Nonparametric estimation o f large covariance matrices of longitudinal data,

    W . B. Wu and M. Pourahmadi, “Nonparametric estimation o f large covariance matrices of longitudinal data,” Biometrika, vol. 90, no. 4, pp. 831–844, 2003

    work page 2003

  38. [38]

    Bregman it erative algorithms for l1-minimization with applica- tions to compressed sensing,

    W . Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman it erative algorithms for l1-minimization with applica- tions to compressed sensing,” SIAM Journal on Imaging sciences , vol. 1, no. 1, pp. 143–168, 2008. 14 A preprint - July 2, 2019

    work page 2008

  39. [39]

    High dimensional inverse covariance matrix e stimation via linear programming,

    M. Y uan, “High dimensional inverse covariance matrix e stimation via linear programming,” Journal of Machine Learning Research, vol. 11, no. Aug, pp. 2261–2286, 2010

    work page 2010

  40. [40]

    Model selection and estimation in the gaussian graphical model,

    M. Y uan and Y . Lin, “Model selection and estimation in the gaussian graphical model,” Biometrika, vol. 94, no. 1, pp. 19–35, 2007

    work page 2007

  41. [41]

    A unified primal-dual algorithm framework based on bregman iteration,

    X. Zhang, M. Burger, and S. Osher, “A unified primal-dual algorithm framework based on bregman iteration,” Journal of Scientific Computing , vol. 46, no. 1, pp. 20–46, 2011

    work page 2011

  42. [42]

    On model selection consistency of las so,

    P . Zhao and B. Y u, “On model selection consistency of las so,” Journal of Machine learning research , vol. 7, no. Nov, pp. 2541–2563, 2006

    work page 2006

  43. [43]

    A greedy algorithm for sparse precision matrix approximation

    H. Zou, “The adaptive lasso and its oracle properties,” Journal of the American statistical association , vol. 101, no. 476, pp. 1418–1429, 2006. 15 This figure "ADMM_200.png" is available in "png" format from: http://arxiv.org/ps/1907.00723v1 This figure "ADMM_EPSILON_200.png" is available in "png" format from: http://arxiv.org/ps/1907.00723v1 This figur...

    work page internal anchor Pith review Pith/arXiv arXiv 2006