Homotopy Based Algorithms for $\ell _{\scriptscriptstyle 0}$-Regularized Least-Squares

Soussen, Charles; Idier, Jérôme; Duan, Junbo; Brie, David

doi:10.1109/tsp.2015.2421476

Cited by 38 publications

(25 citation statements)

References 57 publications

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“…The sparsityconstrained formulation (2) defines a sparse coding problem with a predefined sparsity parameter T . Considering the errorconstrained optimization problem (3), it is easy to make OMP satisfy the constraint by measuring the reconstruction error each time after adding a non-zero entry [7]; The proximal method will search for the Pareto optimal when the sparsity level varies [40]; MIQP keeps all the signals in the constraint based on the decided sparsity of initialization obtained by the proximal method. As recommended in [7], = c n σ 2 with c = 1.15 and a maximum sparsity parameter T m (usually the same as T ) set to assure the sparse level.…”

Section: Large-scale (Global) Dictionary Learningmentioning

confidence: 99%

Mixed Integer Programming For Sparse Coding: Application to Image Denoising

Liu

Canu

Honeiné

et al. 2019

IEEE Trans. Comput. Imaging

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Dictionary learning for sparse representations is generally conducted in two alternating steps: sparse coding and dictionary updating. In this paper, a new approach to solve the sparse coding step is proposed. Because this step involves an 0-norm, most, if not all existing solutions only provide a local or approximate solution. Instead, a real 0 optimization is considered for the sparse coding problem providing a global solution. The proposed method reformulates the optimization problem as a Mixed-Integer Quadratic Program (MIQP), allowing then to obtain the global optimal solution by using an offthe-shelf optimization software. Because computing time is the main disadvantage of this approach, two techniques are proposed to improve its computational speed. One is to add suitable constraints and the other to use an appropriate initialization. The results obtained on an image denoising task demonstrate the feasibility of the MIQP approach for processing real images while achieving good performance compared to the most advanced methods.

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Section: Large-scale (Global) Dictionary Learningmentioning

confidence: 99%

Mixed Integer Programming For Sparse Coding: Application to Image Denoising

Liu

Canu

Honeiné

et al. 2019

IEEE Trans. Comput. Imaging

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“…The stronger (i.e., the more restrictive) the optimality condition verified by points attained by a given algorithm, the "better" the algorithm. Moreover, these conditions can also give rise to new iterative algorithms [2,4,44]. Although a variety of necessary optimality conditions with different degree of sophistication have been defined, analyzed, and hierarchized (see Section 2 and Figure 1), there is a lack of connection between some of them.…”

Section: Necessary Optimality Conditionsmentioning

confidence: 99%

“…By combining the inequalities (43) and (44) with the definition of the CEL0 penalty in (3), we obtain…”

Section: D21 Determination Of ηmentioning

confidence: 99%

New Insights on the Optimality Conditions of the $$\ell _2-\ell _0$$ Minimization Problem

2019

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This paper is devoted to the analysis of necessary (not sufficient) optimality conditions for the 0regularized least-squares minimization problem. Such conditions are the roots of the plethora of algorithms that have been designed to cope with this NP-hard problem. Indeed, as global optimality is, in general, intractable, these algorithms only ensure the convergence to suboptimal points that verify some necessary optimality conditions. The degree of restrictiveness of these conditions is thus directly related to the performance of the algorithms. Within this context, our first goal is to provide a comprehensive review of commonly used necessary optimality conditions as well as known relationships between them. Then, we complete this hierarchy of conditions by proving new inclusion properties between the sets of candidate solutions associated to them. Moreover, we go one step further by providing a quantitative analysis of these sets. Finally, we report the results of a numerical experiment dedicated to the The authors would like to dedicate this work to the memory of Mila Nikolova who passed away in June 2018. Mila made significant contributions to the understanding of the properties of the minimizers of nonconvex regularized least-squares [16, 17, 29-34]. Among them, she published a couple of very instructive papers [31, 32] that provide an in-depth analysis of the minimizers of Problem (1). These works, as well as exiting discussions with Mila herself, have greatly inspired and contributed to the analysis reported in the present paper.

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“…The NMSE is defined as ||x −x|| 2 /||x|| 2 , and the SNR is defined as 20 log(||x −x|| 2 /||x|| 2 ). In order to visually display this performance, we choose the SL0 [12,13], 2 -SL0 [17][18][19] and p -RLS [21] algorithms for comparison.…”

Section: Numerical Simulation and Analysismentioning

confidence: 99%

“…These methods give adequate consideration on sparsity and convergence of the solution, but they are unstable in a noisy environment. Based on this, the 2 -SL0 [17][18][19] transformed the 0 -norm problem into the regularized least squares problem (LSP) [20], which includes sparsity regularizer and deviation term, to improve the performance of sparse vector recovery under noisy conditions. Further, p -RLS [21] that converts the sparsity regularizer into p -norm is introduced to effectively reduce the computational complexity.…”

Section: Introductionmentioning

confidence: 99%

A New Fast and Accurate Compressive Sensing Technique for Magnetic Resonance Imaging

Yue¹,

Yin²

2019

PIER C

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In this paper, the main problem to be solved is how to achieve magnetic resonance imaging (MRI) accurately and quickly. Previous work has shown that compressive sensing (CS) technology can reconstruct a magnetic resonance (MR) image from only a small number of samples, which significantly reduces MR scanning time. Based on this, an algorithm to improve the accuracy of MRI, called regularized weighting Composite Gaussian smoothed 0-norm minimization (RWCGSL0), is proposed in this paper. Different from previous methods, our algorithm has three influential contributions: (1) a new smoothed Composite Gaussian function (CGF) is proposed to be closer to the 0-norm; (2) a new weighting function is proposed; (3) a new 0 regularized objective function framework is constructed. Furthermore, the optimal solution of this objective function is obtained by penalty decomposition (PD) method. It is experimentally shown that the proposed algorithm outperforms other state-of-the-art CS algorithms in the reconstruction of MR images.

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Homotopy Based Algorithms for $\ell _{\scriptscriptstyle 0}$-Regularized Least-Squares

Cited by 38 publications

References 57 publications

Mixed Integer Programming For Sparse Coding: Application to Image Denoising

Mixed Integer Programming For Sparse Coding: Application to Image Denoising

New Insights on the Optimality Conditions of the $$\ell _2-\ell _0$$ Minimization Problem

A New Fast and Accurate Compressive Sensing Technique for Magnetic Resonance Imaging

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