Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization

Ariel Neufeld; Calypso Herrera; Michel Baes; Pierre Ruyssen

REVIEW 3 major objections 4 minor 27 references

Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Neural-net parametrization yields polynomial-in-dimension convergence for low-rank plus sparse matrix decomposition.

desk verdict A useful neural-parametrization trick for low-rank plus sparse decomposition, with an honest but unproven boundedness assumption that keeps the advertised convergence rate conditional. read the letter →

arxiv 1908.00461 v2 pith:ICZDLQKU submitted 2019-08-01 math.OC cs.NAmath.NA

Michel Baes , Calypso Herrera , Ariel Neufeld , Pierre Ruyssen This is my paper · ORCID

classification math.OCcs.NAmath.NA MSC 90C2690C3068T07

keywords low-rankplussparsedecompositioncovariancematrixneuralnetworkparametrizationgradientdescentconvergencenonconvexoptimizationLipschitzsmoothnessportfoliopositivesemidefinite

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 claims that the low-rank plus sparse decomposition of a positive semidefinite matrix can be cast as a smooth optimization problem over the weights of a deep neural network, and that gradient descent on that problem has a proven first-order convergence rate. The low-rank part is written as $L=MM^{T}$, where $M$ is the output of an $m$-layer network, and the loss is a smooth approximation of $\lVert MM^{T}-\Sigma\rVert_1$. The central result is that, on any bounded set of parameters, the gradient of this loss is Lipschitz with a constant that grows only polynomially in the matrix dimension $n$, the rank $k$, and the hidden-layer widths, for fixed depth. Gradient descent therefore reaches a stationary point at rate $O(1/\sqrt{N})$ with a dimension-polynomial constant, assuming the iterates never leave the bounded set. The authors report experiments on synthetic covariance matrices and on correlation matrices of S&P500 stocks and real estate returns, where the method recovers the decomposition accurately, keeps $L$ positive semidefinite by construction, and exhibits bounded parameter sequences.

What carries the argument

The engine of the argument is the pair $(M, \phi)$: the low-rank factor $M=g(N^{\Theta}_m(h(\Sigma)))$, an $n\times k$ matrix produced by a fully connected network with activation $\sigma:\mathbb{R}\to[-1,1]$ having bounded first and second derivatives, and the smoothed $\ell^1$ loss $\phi(\Theta)=\sum_{i,j}\mu([MM^{T}-\Sigma]_{i,j})$. The proof of Theorem II.1 proceeds by first computing explicit partial derivatives for a single hidden layer (Lemmas IV.1 and IV.2), then using a composition-and-product Lipschitz lemma (Lemma IV.3) to bound the smoothness constants of the derivative maps, obtaining the single-layer bound in Lemmas IV.4-IV.6. An induction over the network's $m$ layers then yields the final bound on $L_m$. The smoothing function $\mu$ is what makes the loss differentiable; the boundedness of $\sigma'$ and $\sigma''$ is what keeps the Lipschitz estimates finite and polynomial.

What would settle it

Take a fixed positive semidefinite $\Sigma$, initialize the network once, and run the algorithm with a constant step size while recording the running maximum of $\|\Theta_j\|$; if on some input the running maximum grows without bound, then the boundedness hypothesis behind Theorem II.1 is violated and the rate (14) has no force for that case. A more direct check is to sample two parameter vectors in the same ball, compute the quotient $\|\nabla\phi(\Theta)-\nabla\phi(\Theta')\|/\|\Theta-\Theta'\|$, and compare it with the paper's $L_m$ bound; a value exceeding the bound would contradict the claimed smoothness constant.

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Extended reading notes

Core claim

The paper establishes that a neural-network parametrization turns the rank-constrained low-rank plus sparse decomposition into an optimization problem with a quantitative first-order guarantee. With $M=g(N^{\Theta}_m(h(\Sigma)))\in\mathbb{R}^{n\times k}$ and the smoothed loss $\phi(\Theta)=\sum_{i,j}\mu([MM^{T}-\Sigma]_{i,j})$, where $\mu$ is a smooth approximation of $|t|$ with $|\mu'|\le 1$, Theorem II.1 proves that under the boundedness assumption $\sup_j\|\Theta_j\|\le D$, the gradient $\nabla\phi$ is Lipschitz on that ball with constant $L_m$ whose square satisfies the displayed polynomial bound in $n$, $k$, the layer widths, $D$, and $\|h(\Sigma)\|$. The standard descent argument then yields $\min_{0\le j\le N}\|\nabla\phi(\Theta_j)\|\le \frac{1}{\sqrt{N+1}}\left[\frac{L_m}{K}(\phi(\Theta_0)-\phi^*)\right]^{1/2}$ for any step-size rule satisfying condition (13). This provides a polynomial-in-dimensions first-order convergence rate for a neural-network parametrized decomposition method, and the paper's numerical experiments on synthetic matrices and on two real correlation datasets indicate that the method recovers the intended structure accurately while preserving positive semidefiniteness of $L$.

Load-bearing premise

The load-bearing assumption is that the gradient-descent iterates stay inside a fixed ball $\{\Theta:\|\Theta\|\le D\}$; the paper does not prove this, only checks it numerically on the two real datasets, and if the parameters drift outside the ball the Lipschitz bound and the stated convergence rate can fail.

Editorial extensions

If this is right

A user who fixes the rank $k$ and the network depth can run plain gradient descent with a constant or Armijo step size and be guaranteed to find a point with $\|\nabla\phi\|\le\varepsilon$ in at most $O(L_m/(K\varepsilon^2))$ iterations, where $L_m$ is polynomial in $n$, $k$, and layer widths.
Because $L=MM^{T}$ is always positive semidefinite, the decomposition never produces an indefinite low-rank component, even when the empirical correlation matrix has negative eigenvalues.
Unlike nuclear-norm or PCP-style methods, the rank of $L$ is chosen in advance, so the method directly answers the portfolio problem of fixing the number of economic factors and letting the sparse part reveal residual large correlations.
The explicit dependence of $L_m$ on the parameter bound $D$ and layer widths gives a quantitative tradeoff: smaller parameter bounds and narrower layers improve the rate constant, for fixed depth.
The rate guarantee covers all bounded trajectories of gradient descent; the numerical experiments suggest the boundedness condition holds on realistic correlation matrices, which extends the theoretical guarantee to practice for those cases.

Reading between the lines

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

An implication left implicit is that the same proof template should apply to any factorized model $L=MM^{T}$ with a Lipschitz smooth activation, not only fully connected networks; convolutional or residual parametrizations would need their own compositional Lipschitz estimates.
The convergence guarantee concerns stationarity of the smoothed loss, not recovery of the true $(L_0,S_0)$; connecting the smoothed stationary points to exact decomposition recovery, in the style of robust PCA recovery theorems, is not addressed and may deserve separate analysis.
A practical extension suggested by the bound is to regularize the parameters explicitly (for example, with weight decay) to keep $D$ small, since the rate constant worsens polynomially in $D$; the paper does not analyze such regularization.
Because the loss is just a smooth approximation of $\ell^1$, the method could be extended to weighted or structured sparsity penalties by changing $\mu$ or adding a penalty term, but the convergence proof would require rechecking the second-derivative bound.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know two things about arXiv:1908.00461. First, the genuinely new idea—parametrizing M in L = MM^T by a deep network and running gradient descent on a smoothed l1 loss—is real, and the paper takes it seriously. Second, the advertised convergence rate is conditional on a boundedness assumption that the authors verify only numerically, so the theory is not as unconditional as the abstract suggests.

What the paper does well: the Lipschitz bound computation for the gradient is a genuine technical contribution. They break the single-layer case into lemmas and then induct for multi-layer networks, and the polynomial dependence on dimensions is explicit. The experiments are broader than is typical here: synthetic data with known ground truth, S&P500 correlations, real estate returns, and comparison against four established algorithms. The guarantee that L is positive semidefinite by construction is practically useful, and the code is available. The authors also flag Assumption (12) themselves in Remark II.3, which is more honest than many papers.

The soft spot is exactly what the stress test identifies. Theorem II.1's rate depends on sup_j ||Theta_j|| <= D, and the proof only gives local Lipschitzness on that ball. The running-max plots over a few hundred iterations are suggestive but do not rule out escape on a longer horizon or on other datasets. Worse, L_m in the bound depends on D itself, so the "polynomial in dimensions" claim is really polynomial in dimensions times an unspecified function of D. That matters if you want an a priori complexity estimate. I would also note that the abstract says "convergence rate to a local optimum," but the theorem only bounds the gradient norm of the smoothed surrogate; stationarity of the original l1 objective is not established. That is a real but secondary overstatement.

None of this kills the paper. The proof is detailed, the numerical evidence is mixed but mostly positive, and the conditional nature is disclosed. If the boundedness assumption could be proven for reasonable step-size rules, or if the authors reframed the claim as a rate conditional on bounded iterates, the main objection would go away.

Who gets value from this: people working on nonconvex low-rank recovery or neural parametrization tricks, and finance practitioners who want a rank-controlled decomposition with a PSD guarantee. I would send it to a serious referee. The reviewer's main job should be to push on Assumption (12) and on the mismatch between the smoothed objective and the original problem. I would not desk-reject it; I would also not cite its rate as unconditional until that assumption is resolved.

Referee Report

3 major / 4 minor

Summary. The paper proposes a method for decomposing a positive semidefinite covariance matrix Sigma into a low-rank part L = MM^T plus a sparse part S, where M is parametrized as the output of a fully connected neural network applied to the fixed input h(Sigma). The objective is a smooth approximation of ||MM^T - Sigma||_1, and the authors analyze gradient descent on the network parameters. The main theoretical result, Theorem II.1, gives an O(1/sqrt(N)) bound on the smallest gradient norm of the smoothed objective over N iterations, with a Lipschitz constant that grows polynomially in n, k, and the layer widths for fixed depth m, provided the parameter sequence remains in a fixed ball (Assumption (12)). The proof is carried out through Lemmas IV.1-IV.6, which bound the derivatives and the Lipschitz constant of the smoothed objective for single-layer and multi-layer networks. Numerical experiments on synthetic data, S&P 500 stock correlations, and real estate indices compare the method against IALM, nonconvex RPCA, RPCA-GD, and FPCP. Section III-D plots the running maximum of the parameter norm over 400 and 200 iterations on the two real datasets as empirical evidence for Assumption (12).

Significance. If the unconditional convergence claim were established, the paper would provide a first-order, SVD-free method for low-rank plus sparse covariance decomposition with a polynomial-in-dimensions complexity bound for fixed depth, and the numerical results suggest competitive empirical accuracy. The paper's main strength is the explicit and detailed computation of the Lipschitz constant of the gradient of the smoothed objective, which is a nontrivial compositional calculation; the authors also provide reproducible code and highlight the positive semidefiniteness of L by construction. However, the central theorem is conditional on an unverified boundedness assumption, the rate depends on the unknown bound D, and the convergence guarantee applies only to the smoothed surrogate, not to the original nonsmooth objective. These limitations substantially temper the advertised contribution, so the paper is best viewed as a proof-of-concept plus a worst-case smoothness analysis rather than an unconditional complexity bound.

major comments (3)

[Section II, Theorem II.1 and Remark II.3] The convergence rate (14) is conditional on Assumption (12), sup_j ||Theta_j|| <= D. This assumption is not derived from the gradient dynamics; Remark II.3 concedes that global Lipschitzness of grad(phi) fails and that the gradient grows polynomially, of degree 4 in the single-layer case. The empirical verification in Section III-D is limited to running maxima over 400 (S&P500) and 200 (real estate) iterations on two datasets, which cannot establish an infinite-horizon bound. Moreover, the Lipschitz constant L_m in Theorem II.1 depends on D, with powers such as D^{4m+2}, so the advertised 'polynomial in dimensions' rate is in fact polynomial in dimensions and an unspecified function of D. Please either prove or enforce the boundedness of the iterates, or restate the main result as a conditional guarantee and make the D-dependence explicit in the abstract and the theorem statement.
[Abstract and Theorem II.1] The abstract claims 'convergence rate to a local optimum', but Theorem II.1 only bounds min_{0<=j<=N} ||grad(phi)(Theta_j)|| for the smoothed surrogate phi defined in (9). Since phi is an epsilon-approximation of the nonsmooth l1 objective, a small gradient of phi does not imply stationarity of the original objective ||MM^T - Sigma||_1, let alone convergence to a local optimum of that objective. The authors should state the result as a gradient-norm bound for the smoothed surrogate and explain the relation to the original problem, for example via epsilon-stationarity in the sense of Nesterov's smoothing framework [17].
[Remark II.2, condition (13)] The claim that a constant step-size strategy satisfies (13) with K = 1 is not justified as stated. For an L-smooth function, the standard descent lemma with step h = 1/L gives phi(Theta_j) - phi(Theta_{j+1}) >= (1/(2L))||grad(phi)(Theta_j)||^2, i.e., K = 1/2 in the notation of (13), not K = 1. Please specify the exact step-size rule that yields K = 1, or adjust the numerical constant in Remark II.2.

minor comments (4)

[Section IV, Lemmas IV.4 and IV.5] The lemma statements write 'Theta = (A,d,C,d)', which appears to be a typo; the intended tuple is (A,b,C,d) as used in the proofs and in the single-layer formulation.
[Section III-D and Figure 4] The running-maximum plots in Figure 4 use a very compressed y-axis scale (e.g., 36.76 to 36.81 for S&P500), which makes the visual claim of convergence difficult to assess. A plot of ||Theta_j|| itself or a logarithmic scale would be more informative.
[Section III-A, Table II] For n = 800, the DNN row reports mean relative errors of 1.16 (1.59) for S and 0.32 (0.44) for L with large standard deviations. The text dismisses these as 'outlier instances', but the reader would benefit from a quantitative explanation of when and why these outliers occur.
[Section II, notation around (8)] The norm ||h(Sigma)|| is used without formally defining a norm on the vectorization image of h; it is clear from context that the Euclidean norm is intended, but a brief definition would improve precision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence analysis is self-contained and conditional only on an explicit boundedness assumption; the empirically verified boundedness is not a fitted input.

full rationale

The paper's central claim is Theorem II.1: for a fixed Sigma, if the gradient-descent parameter sequence satisfies sup_j ||Theta_j|| <= D, then the gradient of the smoothed objective phi is Lipschitz on that ball with an explicit polynomial-in-dimensions constant L_m, and standard descent inequalities give the finite-time bound (14). The proof derives L_m from the compositional structure of the network (Lemmas IV.1-IV.6 and the induction in the proof of Theorem II.1) and from standard L-smooth nonconvex optimization theory; it does not use any fitted parameter, no normalization is calibrated to Sigma, and no prior result by the same authors is invoked. The only additional assumption is (12), which is explicitly stated and not derived from the dynamics; Remark II.3 even concedes that global Lipschitzness fails and that the gradient grows polynomially in the parameters. The empirical running-max checks in Section III-D support (12) on two data sets, but they are not part of the proof and do not make the theorem circular. The abstract's phrase 'convergence rate to a local optimum' overstates the gradient-norm bound in (14), but that is a correctness or interpretation gap, not circularity. No self-citation is load-bearing: the reference list contains no work by Baes, Herrera, Neufeld, or Ruyssen. Accordingly, the derivation is self-contained and the circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The theoretical result has no fitted parameters; the method's user-chosen hyperparameters (smoothing epsilon, rank k, network architecture, step sizes) and the unverified boundedness assumption D are listed. No new physical entities are introduced.

free parameters (5)

smoothing parameter epsilon
Appears in the smooth absolute value approximation mu(t) in Section II; chosen as a small positive constant, not fitted by theory. Affects the fidelity of the l1 loss approximation.
rank k
User-selected upper bound on the rank of L; the algorithm is run for a given k, and the results depend on its choice.
network architecture (depth m, widths l_1,...,l_m, activation sigma)
Chosen by the user; the paper reports that variations only marginally influence the numerical results. The Lipschitz bound scales polynomially in these quantities for fixed depth.
step-size sequence h_j
Must satisfy condition (13), e.g., constant step size or Goldstein-Armijo rule. The actual values are not specified, though the convergence result depends on the condition.
boundedness constant D
Assumption (12) in Theorem II.1 requires sup_j ||Theta_j|| <= D. The value is not known a priori; the authors verify empirically that parameters stay bounded but do not provide a theoretical bound.

assumptions (6)

domain assumption Activation function sigma: R -> [-1,1] with uniformly bounded first and second derivatives.
Used throughout Section II and the proof in Section IV; satisfied by tanh, logistic, and scaled arctan (Table I).
domain assumption The parameter sequence (Theta_j) stays in a bounded set (Assumption (12)).
Critical for the Lipschitz bound in Theorem II.1; empirically verified in Section III-D but not proven.
domain assumption The step-size strategy satisfies the sufficient decrease condition (13).
Standard in gradient descent theory; shown in Remark II.2 to hold for constant step size and Armijo-type rules.
standard math The smooth approximation mu has derivative bounded by 1 and second derivative bounded by mu''_max.
Used to define the differentiable objective (9); examples given in Section II.
standard math Any rank-k positive semidefinite matrix L can be factored as M M^T with M in R^{n x k}.
Facts from linear algebra used in Section II to represent the low-rank part.
standard math For L-smooth nonconvex functions, gradient descent with sufficient decrease converges to a first-order stationary point at rate O(1/N).
Classical result (Nesterov [18], Ghadimi-Lan [8]) used in the second part of Theorem II.1.

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Pith. "Pith review of Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization." pith.science (2026). https://pith.science/paper/ICZDLQKU

@misc{pith2026190800461,
  author       = {Pith},
  title        = {Pith review of: Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICZDLQKU}},
  note         = {Machine review of arXiv:1908.00461}
}

abstract

This paper revisits the problem of decomposing a positive semidefinite matrix as a sum of a matrix with a given rank plus a sparse matrix. An immediate application can be found in portfolio optimization, when the matrix to be decomposed is the covariance between the different assets in the portfolio. Our approach consists in representing the low-rank part of the solution as the product $MM^{T}$, where $M$ is a rectangular matrix of appropriate size, parametrized by the coefficients of a deep neural network. We then use a gradient descent algorithm to minimize an appropriate loss function over the parameters of the network. We deduce its convergence rate to a local optimum from the Lipschitz smoothness of our loss function. We show that the rate of convergence grows polynomially in the dimensions of the input, output, and the size of each of the hidden layers.

Figures

Figures reproduced from arXiv: 1908.00461 by the authors.

**Figure 2.** Decomposition into a low-rank plus a sparse matrix [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The first 17 (left panels) and 50 (right panels) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The running maximum max0≤j≤J kΘjk (y-axis) of our parameters (Θj )j∈N0 generated by our DNN algorithm, plotted as a function of iterations J (x-axis) for both examples on the S&P500 stock prices (left) and on the real estate returns (right). For establishing this boun…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 25 canonical work pages

[17]

Nesterov

Y . Nesterov. Smooth minimization of non-smooth functions. Mathematical programming, 103(1):127–152, 2005

work page 2005
[1]

Abadi, A

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y . Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Man ´e, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V . Vanhouck...

work page 2015
[2]

Aybat, L

N. Aybat, L. Xue, and H. Zou. Alternating direction methods for latent variable gaussian graphical model selection. Neural Computation, 25:2172–2198, 2013

work page 2013
[3]

Bouwmans, N

T. Bouwmans, N. Aybat, and E. Zahzah. Handbook of Robust Low- Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing. Chapman & Hall/CRC, 2016

work page 2016
[4]

Bouwmans, A

T. Bouwmans, A. Sobral, S. Javed, S. Jung, and E. Zahzah. Decomposition into low-rank plus additive matrices for back- ground/foreground separation: A review for a comparative evalua- tion with a large-scale dataset. Computer Science Review, 23:1–71, 2017

work page 2017
[5]

Bouwmans, A

T. Bouwmans, A. Sobral, and E. Zahzah. Lrslibrary: Low-rank and sparse tools for background modeling and subtraction in videos. In Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing . CRC Press, Taylor and Francis Group, 2015

work page 2015
[6]

Cand `es, X

E. Cand `es, X. Li, Y . Ma, and J. Wright. Robust principal component analysis? Journal of the ACM (JACM), 58(3):11, 2011

work page 2011
[7]

Chandrasekaran, P

V . Chandrasekaran, P. Parrilo, and A. Willsky. Latent variable graphical model selection via convex optimization. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1610–1613. IEEE, 2010

work page 2010

Show all 27 references

[8]

Ghadimi and G

S. Ghadimi and G. Lan. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Mathematical Programming, 156(1-2):59–99, 2016

2016
[9]

J. He, L. Balzano, and A. Szlam. Incremental gradient on the grassmannian for online foreground and background separation in subsampled video. In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pages 1568–1575, 2012

2012
[10]

Hestenes

M. Hestenes. Multiplier and gradient methods. Journal of Optimization Theory and Applications , 4(5):303–320, 1969

1969
[11]

Higham and N

N. Higham and N. Strabic. Bounds for the distance to the nearest correlation matrix. SIAM Journal on Matrix Analysis and Applications, 37(3):1088–1102, 2016

2016
[12]

Z. Kang, C. Peng, Q. Cheng. Robust PCA Via Nonconvex Rank Approximation. In 2015 IEEE International Conference on Data Mining, pages 211–220

2015
[13]

Z. Lin, M. Chen, and Y . Ma. The augmented Lagrange multi- plier method for exact recovery of corrupted low-rank matrices. arXiv:1009.5055, 2010

2010 arXiv
[14]

J. Liu, Y . Xu, C. Zheng, H. Kong, and Z. Lai. RPCA-based tumor classiﬁcation using gene expression data. IEEE/ACM Transactions on Computational Biology and Bioinformatics , 12(4):964–970, 2015

2015
[15]

Lopez-Paz and L

D. Lopez-Paz and L. Sagun. Easing non-convex optimization with neural networks. https:// openreview.net/forum?id=rJXIPK1PM, 2018

2018
[16]

K. Min, Z. Zhang, J. Wright, and Y . Ma. Decomposing background topics from keywords by principal component pursuit. In Proceed- ings of the 19th ACM international conference on Information and knowledge management, pages 269–278. ACM, 2010

2010
[18]

Nesterov

Y . Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013

2013
[19]

Nocedal and S

J. Nocedal and S. Wright. Numerical Optimization. Springer, 2000

2000
[20]

Paszke, S

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. Automatic differentiation in Pytorch. NIPS 2017 Autodiff Workshop , 2017

2017
[21]

Rodriguez and B

P. Rodriguez and B. Wohlberg. Fast principal component pursuit via alternating minimization. In 2013 IEEE International Confer- ence on Image Processing , pages 69–73, 2013

2013
[22]

Rodriguez and B

P. Rodriguez and B. Wohlberg. Incremental principal component pursuit for video background modeling. Journal of Mathematical Imaging and Vision, 55(1):1–18, 2016

2016
[23]

Shahid, V

N. Shahid, V . Kalofolias, X. Bresson, M. Bronstein, and P. Van- dergheynst. Robust Principal Component Analysis on graphs. In Proceedings of the IEEE International Conference on Computer Vision, pages 2812–2820, 2015

2015
[24]

Shkolnik, L

A. Shkolnik, L. Goldberg, and J. Bohn. Identifying broad and narrow ﬁnancial risk factors with convex optimization. SSRN 2800237, 2016

2016
[25]

N. Wang, T. Yao, J. Wang, and D. Yeung. A probabilistic approach to robust matrix factorization. In European Conference on Computer Vision , pages 126–139. Springer, 2012

2012
[26]

X. Zhou, C. Yang, and W. Yu. Moving object detection by detecting contiguous outliers in the low-rank representation. IEEE Transac- tions on pattern analysis and machine intelligence , 35(3):597–610, 2013

2013
[27]

X. Yi, D. Park, Y . Chen, and C. Caramanis. Fast Algorithms for Robust PCA via Gradient Descent. Advances in Neural Information Processing Systems 29 , 4152–4160, 2016

2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Nesterov

Y . Nesterov. Smooth minimization of non-smooth functions. Mathematical programming, 103(1):127–152, 2005

work page 2005

[2] [1]

Abadi, A

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y . Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Man ´e, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V . Vanhouck...

work page 2015

[3] [2]

Aybat, L

N. Aybat, L. Xue, and H. Zou. Alternating direction methods for latent variable gaussian graphical model selection. Neural Computation, 25:2172–2198, 2013

work page 2013

[4] [3]

Bouwmans, N

T. Bouwmans, N. Aybat, and E. Zahzah. Handbook of Robust Low- Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing. Chapman & Hall/CRC, 2016

work page 2016

[5] [4]

Bouwmans, A

T. Bouwmans, A. Sobral, S. Javed, S. Jung, and E. Zahzah. Decomposition into low-rank plus additive matrices for back- ground/foreground separation: A review for a comparative evalua- tion with a large-scale dataset. Computer Science Review, 23:1–71, 2017

work page 2017

[6] [5]

Bouwmans, A

T. Bouwmans, A. Sobral, and E. Zahzah. Lrslibrary: Low-rank and sparse tools for background modeling and subtraction in videos. In Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing . CRC Press, Taylor and Francis Group, 2015

work page 2015

[7] [6]

Cand `es, X

E. Cand `es, X. Li, Y . Ma, and J. Wright. Robust principal component analysis? Journal of the ACM (JACM), 58(3):11, 2011

work page 2011

[8] [7]

Chandrasekaran, P

V . Chandrasekaran, P. Parrilo, and A. Willsky. Latent variable graphical model selection via convex optimization. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1610–1613. IEEE, 2010

work page 2010