Strongly convex programming for exact matrix completion and robust principal component analysis

Zhang, Hui; Cai, Jian-Feng; Cheng, Lizhi; Zhu, Jun

doi:10.3934/ipi.2012.6.357

Cited by 21 publications

(27 citation statements)

References 45 publications

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“…It has been shown in [47,76] that the SVT algorithm with a finite τ can get the perfect matrix completion as (2) does. The SVT algorithm is reformulated as Uzawa's algorithm [4] or linearized Bregman iteration [9,10,58,73].…”

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confidence: 99%

Fast singular value thresholding without singular value decomposition

Cai

Osher

2013

Methods and Applications of Analysis

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Abstract. Singular value thresholding (SVT) is a basic subroutine in many popular numerical schemes for solving nuclear norm minimization that arises from low-rank matrix recovery problems such as matrix completion. The conventional approach for SVT is first to find the singular value decomposition (SVD) and then to shrink the singular values. However, such an approach is time-consuming under some circumstances, especially when the rank of the resulting matrix is not significantly low compared to its dimension. In this paper, we propose a fast algorithm for directly computing SVT for general dense matrices without using SVDs. Our algorithm is based on matrix Newton iteration for matrix functions, and the convergence is theoretically guaranteed. Numerical experiments show that our proposed algorithm is more efficient than the SVD-based approaches for general dense matrices.

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confidence: 99%

Fast singular value thresholding without singular value decomposition

Cai

Osher

2013

Methods and Applications of Analysis

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“…Each solution to the dual problem (21) can generate the unique solution to the primal problem (5) via formulation (20). This fact is stated in the following lemma.…”

Section: Lagrange Dual Analysismentioning

confidence: 96%

Projected shrinkage algorithm for box-constrained $$\ell _1$$ ℓ 1 -minimization

Zhang

Cheng

2015

Optim Lett

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Box-constrained 1 -minimization in some cases performs remarkably better than the classical 1 -minimization when appropriate box constraints are available. And also many practical 1 -minimization models indeed involve box constraints. In this paper, we propose an efficient iteration scheme, dubbed the projected shrinkage (ProShrink) algorithm, to solve a class of box-constrained 1 -minimization problems.A key component in our technique is that the proximal point operator of 1 -norm with box constraints can be equivalently simplified into a projected shrinkage operator which can be calculated directly. Theoretically, we prove that ProShrink enjoys convergence of both the primal and dual point sequences. On the numerical level, we demonstrate the benefit of adding box constraints via sparse recovery experiments.

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“…For example, one can construct an example of nuclear norm minimization that does not have the exact penalty property. Results are in [59], as well as [95] (and the correction [94]), which also provides results for the RPCA problem in particular. Research in this is motivated by the popularity of the [25] algorithm, which is a special case of the TFOCS framework applied to matrix completion.…”

Section: Effect Of the Smoothing Termmentioning

confidence: 98%