Max-norm optimization for robust matrix recovery

Fang, Ethan X.; Liu, Han; Toh, Kim Chuan; Zhou, Wen‐Xin

doi:10.1007/s10107-017-1159-y

Cited by 18 publications

(27 citation statements)

References 32 publications

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“…Therefore, their algorithm is only guaranteed to find a stationary point, and statistical properties of such solutions are difficult to analyze. Recently, Fang et al (2015b) proposed a scalable algorithm based on the alternating direction of multipliers method to efficiently solve the max-norm constrained optimization problem with guaranteed rate of convergence to the global optimum. In summary, the max-norm constrained empirical risk minimization problem can indeed be implemented in polynomial time as a function of the sample size and matrix dimensions.…”

Section: Introductionmentioning

confidence: 99%

Matrix completion via max-norm constrained optimization

Cai¹,

Zhou²

2016

Electron. J. Statist.

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Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform.In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.MSC 2010 subject classifications: Primary 62H12, 62J99; secondary 15A83.

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Section: Introductionmentioning

confidence: 99%

Matrix completion via max-norm constrained optimization

Cai¹,

Zhou²

2016

Electron. J. Statist.

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“…Passing the limit ν → ∞ to the both sides and using Lemma 3.2 yields ∞ j=k x j+1 −x j < ∞. By combining the last inequality with (12) and recalling the definition of c 1 (θ) in (18), we obtain 4m+3 4(m+1)…”

mentioning

confidence: 88%

“…and that the samples of the indices are drawn independently from a general sampling distribution Π = {π kl } k∈[n1],l∈[n2] on [n 1 ] × [n 2 ]. We adopt the same non-uniform sampling scheme as in [18], i.e., for each (k, l) ∈ [n 1 ] × [n 2 ], take π kl = p k p l with p k = 2p 0 if k ≤ n1 10 , p k = 4p 0 if n1 10 ≤ k ≤ n1 5 , otherwise p k = p 0 , where p 0 > 0 is a constant such that n1 k=1 p k = 1, and p l is defined in a similar way. The entries M it,jt with (i t , j t ) ∈ Ω for t = 1, 2, .…”

Section: Algorithm 2 (Nonmonotone Line Search Palm With Extrapolation)mentioning

confidence: 99%

Convergence of a class of nonmonotone descent methods for KL optimization problems

Qian¹,

Pan²

2022

Preprint

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This paper is concerned with a class of nonmonotone descent methods for minimizing a proper lower semicontinuous KL function Φ, which generates a sequence satisfying a nonmonotone decrease condition and a relative error tolerance. Under mild assumptions, we prove that the whole sequence converges to a limiting critical point of Φ and, when Φ is a KL function of exponent θ ∈ [0, 1), the convergence admits a linear rate if θ ∈ [0, 1/2] and a sublinear rate associated to θ if θ ∈ (1/2, 1). The required assumptions are shown to be necessary when Φ is weakly convex but redundant when Φ is convex. Our convergence results resolve the convergence problem on the iterate sequence generated by nonmonotone line search algorithms for nonconvex nonsmooth problems, and also extend the convergence results of monotone descent methods for KL optimization problems. As the applications, we achieve the convergence of the iterate sequence for the nonmonotone line search proximal gradient method with extrapolation and the nonmonotone line search proximal alternating minimization method with extrapolation. Numerical experiments are conducted for zero-norm and column ℓ 2,0 -norm regularized problems to validate their efficiency.

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“…The max-norm is an alternative convex relaxation for the rank in the low-complexity matrix completion problem, which is considered to be better than nuclear-norm in theoretical cases [52], [59]. The data missing spectral matrix completion based on max-norm can be formulated as the following constrained optimization problem:…”

Section: Max-norm Minimizationmentioning

confidence: 99%

Low Complexity Modeling of Cross-Spectral Matrix and Its Application in the Non-Synchronous Measurements of Microphones Array

Fan

Chu

et al. 2021

IEEE Access

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The resolution related with the image quality of acoustic imaging using a microphone array is limited by the size and density of the array. However, non-synchronous measurements can exceed the constraints defined by measurements with a single fixed array. The non-synchronous measurements can achieve a larger and denser aperture array by moving the single array. The key problem of archiving the non-synchronous measurements of a microphone array is the matrix completion of a block diagonal spectral matrix. A comprehensive investigation of the block diagonal spectral matrix completion is still missing in the microphone array signal processing literature since it is usually considered as a difficult problem if without the integrated physical information of the sound field. In this paper, the mathematical relation between the max-norm and the nuclear-norm is pointed out, and the max-norm minimization of the spectral matrix is discussed, which is an effective model with integrated spatial representation of the sound field for the spectral matrix completion. First, the max-norm is utilized for the low complexity modeling and the corresponding Proximal-Point method for Max-norm based Spectral matrix completion (PPMSMC) is proposed; Second, the rank, nuclear-norm and max-norm have been investigated as complexity measures in the context of spectral matrix completion, and their performances are compared. The proposed low-rank methods have constructed a concrete theorem foundation for the non-synchronous measurements of the microphone array.INDEX TERMS Acoustic measurements, low complexity model, max-norm minimization.

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Max-norm optimization for robust matrix recovery

Cited by 18 publications

References 32 publications

Matrix completion via max-norm constrained optimization

Matrix completion via max-norm constrained optimization

Convergence of a class of nonmonotone descent methods for KL optimization problems

Low Complexity Modeling of Cross-Spectral Matrix and Its Application in the Non-Synchronous Measurements of Microphones Array

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