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arxiv: 2507.17069 · v3 · submitted 2025-07-22 · 🧮 math.OC · cs.NA· math.NA

The Generalized Matrix Separation Problem: Algorithms

Xuemei Chen , Owen Deen This is my paper

Pith reviewed 2026-05-19 02:49 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords matrix separationlow-rank matrixsparse matrixpreconditioningconvex optimizationiterative algorithmsnuclear norml1 norm
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The pith

Preconditioning enables efficient iterative algorithms for generalized matrix separation.

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

This paper develops practical algorithms to solve a convex optimization problem that recovers a low-rank matrix and a sparse matrix from an observation formed by their sum after a linear transformation by H. It provides detailed iterative procedures and focuses on efficient computations when H has circulant, separable, or block structure. A key contribution is a preconditioning technique shown to improve efficiency, accuracy, and robustness, backed by theoretical guarantees. Readers interested in computational methods for matrix decomposition would find this useful for turning abstract models into implementable code.

Core claim

We present iterative algorithms for solving the convex program that minimizes the nuclear norm of L plus the l1 norm of S subject to M = L + H S. Implementation strategies are given for practical structures of H, and a preconditioning technique is proposed that drastically improves performance in efficiency, accuracy, and robustness, with a theoretical guarantee provided for the strategy.

What carries the argument

The preconditioning technique for the iterative solvers that exploits structures in H to speed up and stabilize the separation of low-rank and sparse components.

If this is right

  • The proposed algorithms become practical for structured H through efficient matrix operations.
  • Preconditioning leads to better accuracy and robustness in recovering the matrices.
  • Theoretical guarantees support the reliability of the preconditioned methods.
  • Numerical results confirm the effectiveness for the generalized separation task.

Where Pith is reading between the lines

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

  • This could connect to fast Fourier transform based methods when H is circulant.
  • Extensions to other convex problems with nuclear and l1 norms may benefit from similar preconditioning.
  • Applications in areas like robust PCA with transformations could see practical gains.

Load-bearing premise

The preconditioning strategy depends on H having exploitable structures such as circulant or block forms that allow for efficient operations and effective preconditioner design.

What would settle it

A direct comparison showing that the preconditioned algorithms perform no better than standard methods on problems where H lacks the assumed structures would challenge the claims.

Figures

Figures reproduced from arXiv: 2507.17069 by Owen Deen, Xuemei Chen.

Figure 1
Figure 1. Figure 1: Convergence of Algorithm 3 (no preconditioning) vs. Algorithm 4 (with preconditioning). [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Preconditioning (right figure) allows more robust and wider range of choices of [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Recoverability results when H is Gaussian 19 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Recoverability results when H is circulant (deterministic) 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simultaneous background removal and deblurring [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

When given a generalized matrix separation problem, which aims to recover a low rank matrix $L_0$ and a sparse matrix $S_0$ from $M_0=L_0+HS_0$, the work \cite{CW25} proposes a novel convex optimization problem whose objective function is the sum of the $\ell_1$-norm and nuclear norm. In this paper we detail the iterative algorithms and its associated computations for solving this convex optimization problem. We present various efficient implementation strategies, with attention to practical cases where $H$ is circulant, separable, or block structured. Notably, we propose a preconditioning technique that drastically improved the performance of our algorithms in terms of efficiency, accuracy, and robustness. While this paper serves as an illustrative algorithm implementation manual, we also provide theoretical guarantee for our preconditioning strategy. Numerical results demonstrate the effectiveness of the proposed approach.

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

1 major / 1 minor

Summary. The paper presents iterative algorithms for the convex optimization problem of minimizing the nuclear norm of L plus the ℓ1-norm of S subject to the constraint M = L + H S, extending the formulation from prior work CW25. It details efficient implementation strategies specifically for cases where the operator H is circulant (leveraging FFT), separable, or block-structured. A preconditioning technique is proposed that is claimed to substantially improve efficiency, accuracy, and robustness, accompanied by a theoretical guarantee; numerical experiments are included to demonstrate performance.

Significance. If the derivations, convergence analysis, and preconditioning bound hold under the stated structural assumptions on H, the manuscript supplies a practical algorithmic toolkit and implementation guide for generalized matrix separation. The explicit treatment of structured operators (circulant via FFT, separable, block) and the inclusion of numerical validation constitute strengths, as does the focus on reproducible computational details for these common cases in applications such as imaging or signal processing.

major comments (1)
  1. [Preconditioning analysis and abstract] The theoretical guarantee for the preconditioning strategy (referenced in the abstract and developed in the implementation sections) is tied to H possessing exploitable structure such as circulant or separable form that permits fast diagonalization or Kronecker-product identities. The manuscript should explicitly delineate the precise conditions under which the contraction-rate or iteration-complexity bound holds and confirm that no claim is made for unstructured general operators; otherwise the abstract's unqualified statement of a 'theoretical guarantee' risks overstatement.
minor comments (1)
  1. [Abstract and references] The citation to CW25 in the abstract and introduction should be expanded to a full bibliographic entry in the references section to aid readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. The recommendation for minor revision is appreciated, and we address the major comment below by agreeing to strengthen the clarity of the manuscript.

read point-by-point responses

  1. Referee: [Preconditioning analysis and abstract] The theoretical guarantee for the preconditioning strategy (referenced in the abstract and developed in the implementation sections) is tied to H possessing exploitable structure such as circulant or separable form that permits fast diagonalization or Kronecker-product identities. The manuscript should explicitly delineate the precise conditions under which the contraction-rate or iteration-complexity bound holds and confirm that no claim is made for unstructured general operators; otherwise the abstract's unqualified statement of a 'theoretical guarantee' risks overstatement.

    Authors: We agree that the abstract's reference to a 'theoretical guarantee' for the preconditioning strategy should be qualified to avoid any potential overstatement. The analysis and bounds in the implementation sections are derived under the assumption that H has exploitable structure (circulant via FFT, separable, or block-structured) that enables fast diagonalization or Kronecker-product representations. In the revised manuscript, we will explicitly delineate these precise conditions in the abstract and relevant sections, and we will add a clarifying statement confirming that no such guarantee is claimed for unstructured general operators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithms and preconditioning are independent of the cited formulation

full rationale

The paper explicitly attributes the convex optimization problem (nuclear norm plus l1 norm for recovering L0 and S0 from M0 = L0 + H S0) to the prior work cited as CW25, then develops separate iterative algorithms, structure-specific implementations (circulant via FFT, separable, block), and a new preconditioning technique together with its theoretical guarantee. These contributions are presented as implementation details and performance improvements with numerical validation; none of the described steps reduce the new material to the input formulation by definition or by a self-citation chain that itself lacks independent support. The preconditioning analysis is tied to exploitable structure in H but is not shown to be tautological with the original objective.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the convex formulation from prior work and standard assumptions in iterative optimization methods for nuclear and l1 norms; no new free parameters, axioms beyond domain standards, or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Iterative proximal or splitting methods can efficiently solve the sum of nuclear norm and l1 norm objective under linear constraints involving H.
    Standard assumption drawn from convex optimization literature for matrix separation problems.

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 1 internal anchor

  1. [1]

    1, 183–202

    Amir Beck and Marc Teboulle,A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM journal on imaging sciences2(2009), no. 1, 183–202

    work page 2009

  2. [2]

    1, 1–122

    Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein,Distributed optimization and statistical learning via the alternating direction method of multipliers, Foun- dations and Trends®in Machine Learning3(2011), no. 1, 1–122

    work page 2011

  3. [3]

    20, 1–33

    HanQin Cai, Jian-Feng Cai, and Ke Wei,Accelerated alternating projections for robust principal component analysis, Journal of Machine Learning Research20(2019), no. 20, 1–33

    work page 2019

  4. [4]

    HanQin Cai, Keaton Hamm, Longxiu Huang, Jiaqi Li, and Tao Wang,Rapid robust principal component analysis: Cur accelerated inexact low rank estimation, IEEE Signal Processing Letters28(2020), 116–120

    work page 2020

  5. [5]

    4, 1956–1982

    Jian-Feng Cai, Emmanuel J Cand` es, and Zuowei Shen,A singular value thresholding algorithm for matrix completion, SIAM Journal on optimization20(2010), no. 4, 1956–1982

    work page 2010

  6. [6]

    Emmanuel J Cand` es, Xiaodong Li, Yi Ma, and John Wright,Robust principal component analysis?, Journal of the ACM (JACM)58(2011), no. 3, 1–37

    work page 2011

  7. [7]

    2, 572– 596

    Venkat Chandrasekaran, Sujay Sanghavi, Pablo A Parrilo, and Alan S Willsky,Rank-sparsity incoherence for matrix decomposition, SIAM Journal on Optimization21(2011), no. 2, 572– 596

    work page 2011

  8. [8]

    Xuemei Chen and Rongrong Wang,A masked matrix separation problem: A first analyais, arXiv:2504.19025

    work page arXiv

  9. [9]

    11, 1413–1457

    Ingrid Daubechies, Michel Defrise, and Christine De Mol,An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences57(2004), no. 11, 1413–1457. 21

    work page 2004

  10. [10]

    Wei Deng and Wotao Yin,On the global and linear convergence of the generalized alternating direction method of multipliers, Journal of Scientific Computing66(2016), 889–916

    work page 2016

  11. [11]

    33, 2019, pp

    Aritra Dutta, Filip Hanzely, and Peter Richt´ arik,A nonconvex projection method for robust pca, Proceedings of the AAAI conference on artificial intelligence, vol. 33, 2019, pp. 1468–1476

    work page 2019

  12. [12]

    8, 906–916

    M´ ario AT Figueiredo and Robert D Nowak,An em algorithm for wavelet-based image restora- tion, IEEE Transactions on Image Processing12(2003), no. 8, 906–916

    work page 2003

  13. [13]

    4, 586–597

    M´ ario AT Figueiredo, Robert D Nowak, and Stephen J Wright,Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE Journal of selected topics in signal processing1(2008), no. 4, 586–597

    work page 2008

  14. [14]

    1, 17–40

    Daniel Gabay and Bertrand Mercier,A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & mathematics with applications2 (1976), no. 1, 17–40

    work page 1976

  15. [15]

    3, 644–658

    Euhanna Ghadimi, Andre Teixeira, Iman Shames, and Karl Henrik Johansson,Optimal param- eter selection for the alternating direction method of multipliers (admm): Quadratic problems, IEEE Transactions on Automatic Control60(2014), no. 3, 644–658

    work page 2014

  16. [16]

    Analyse num´ erique9 (1975), no

    Roland Glowinski and Americo Marroco,Sur l’approximation, par ´ el´ ements finis d’ordre un, et la r´ esolution, par p´ enalisation-dualit´ e d’une classe de probl` emes de dirichlet non lin´ eaires, Revue fran¸ caise d’automatique, informatique, recherche op´ erationnelle. Analyse num´ erique9 (1975), no. R2, 41–76

    work page 1975

  17. [17]

    Trevor Hastie, Robert Tibshirani, and Jerome Friedman,The elements of statistical learning: Data mining, inference, and prediction, 2 ed., Springer, New York, 2009

    work page 2009

  18. [18]

    1, 148–172

    Misha E Kilmer, Karen Braman, Ning Hao, and Randy C Hoover,Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM Journal on Matrix Analysis and Applications34(2013), no. 1, 148–172

    work page 2013

  19. [19]

    Zhouchen Lin, Minming Chen, and Yi Ma,The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices, arXiv preprint arXiv:1009.5055 (2010)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [20]

    Praneeth Netrapalli, Niranjan UN, Sujay Sanghavi, Animashree Anandkumar, and Prateek Jain,Non-convex robust pca, Advances in neural information processing systems27(2014)

    work page 2014

  21. [21]

    Robert Nishihara, Laurent Lessard, Ben Recht, Andrew Packard, and Michael Jordan,A general analysis of the convergence of admm, International conference on machine learning, PMLR, 2015, pp. 343–352

    work page 2015

  22. [22]

    Antoine Vacavant, Thierry Chateau, Alexis Wilhelm, and Laurent Lequievre,A benchmark dataset for outdoor foreground/background extraction, Asian Conference on Computer Vision, Springer, 2012, pp. 291–300

    work page 2012

  23. [23]

    Xiangfeng Wang, Mingyi Hong, Shiqian Ma, and Zhi-Quan Luo,Solving multiple-block separa- ble convex minimization problems using two-block alternating direction method of multipliers, Pacific Journal of Optimization11(2015), 645–667

    work page 2015

  24. [24]

    Xiaoming Yuan and Junfeng Yang,Sparse and low rank matrix decomposition via alternating direction method, Pacific Journal of Optimization9(2009). 22 (a) original (unknown) (b) blurred (givenM 0) (c) recovered moving objects - FISTA (d) recovered background - FISTA (e) recovered moving objects - ADMM (f) recovered background- ADMM Figure 5: Simultaneous ba...

    work page 2009