$$S_{1/2}$$ S 1 / 2 regularization methods and fixed point algorithms for affine rank minimization problems

Peng, Dingtao; Xiu, Naihua; Yu, Jian

doi:10.1007/s10589-017-9898-5

Cited by 22 publications

(12 citation statements)

References 36 publications

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“…⊤ is an optimal solution to (29), and c j is positive value and μ satisfies 0 < μ ≤ ‖X * ‖ − 2 2 , and then, the optimal solution β * ji can be given by…”

Section: Lemmamentioning

confidence: 99%

Sparse Principal Component Analysis via Fractional Function Regularity

Han

Peng

Cui

et al. 2020

Mathematical Problems in Engineering

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In this paper, we describe a novel approach to sparse principal component analysis (SPCA) via a nonconvex sparsity-inducing fraction penalty function SPCA (FP-SPCA). Firstly, SPCA is reformulated as a fraction penalty regression problem model. Secondly, an algorithm corresponding to the model is proposed and the convergence of the algorithm is guaranteed. Finally, numerical experiments were carried out on a synthetic data set, and the experimental results show that the FP-SPCA method is more adaptable and has a better performance in the tradeoff between sparsity and explainable variance than SPCA.

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“…⊤ is an optimal solution to (29), and c j is positive value and μ satisfies 0 < μ ≤ ‖X * ‖ − 2 2 , and then, the optimal solution β * ji can be given by…”

Section: Lemmamentioning

confidence: 99%

Sparse Principal Component Analysis via Fractional Function Regularity

Han

Peng

Cui

et al. 2020

Mathematical Problems in Engineering

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“…Its roots (complex or real) and their computational formulae have a long history with fascinating and entertaining stories. A comprehensive revisit of this subject can be found in Xing [51] and a successful application of the depressed cubic equation to the compressed sensing can be found in [36,52]. The following lemma says that, under certain conditions, the Eq.…”

Section: Positive Roots Of Depressed Cubic Equationsmentioning

confidence: 99%

Robust Euclidean embedding via EDM optimization

Zhou

Xiu

2019

Math. Prog. Comp.

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This paper aims to propose an efficient numerical method for the most challenging problem known as the robust Euclidean embedding (REE) in the family of multidimensional scaling (MDS). The problem is notoriously known to be nonsmooth, nonconvex and its objective is non-Lipschitzian. We first explain that the semidefinite programming (SDP) relaxations and Euclidean distance matrix (EDM) approach, popular for other types of problems in the MDS family, failed to provide a viable method for this problem. We then propose a penalized REE (PREE), which can be economically majorized. We show that the majorized problem is convex provided that the penalty parameter is above certain threshold. Moreover, it has a closed-form solution, resulting in an efficient algorithm dubbed as PREEEDM (for Penalized REE via EDM optimization). We prove among others that PREEEDM converges to a stationary point of PREE, which is also an approximate critical point of REE. Finally, the efficiency of PREEEDM is compared with several state-of-the-art methods including SDP and EDM solvers on a large number of test problems from sensor network localization and molecular conformation.

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“…Peng et al [46] have shown that the q (0 < q < 1) norm is efficient for the lowrank matrix recovery problem. Considering the q (0 < q < 1) norm for the LLR problem, two questions should be answered.…”

Section: A 1/2 Regularization-based Relaxation Of Rank Minimization mentioning

confidence: 99%

“…It is noteworthy to point out that the rank of the mean filtering matrix is always equal to 1 under the LRR framework (the minimum rank to be reached), whereas the rank obtained by LRR method is usually higher than the real one. Since the proposed LhalfLRR method can achieve lower rank than LRR (as shown in the experiment section), and the LhalfLRR model is more robust than LRR model on noise [46]- [49], it can be a better choice for spatial information fusion of hyperspectral imagery. Through applying the proposed LhalfLRR approach on each pixel of hyperspectral imagery, the spatial structure information can be intelligently incorporated into the center and eventually a processed data cube R * can be obtained (only the center pixel of S is reserved for reconstruction while the rest pixels are omitted, and the size of R * is the same as the original hyperspectral data cube R).…”

Section: B Solving the 1/2 Regularized Optimization Problemmentioning

confidence: 99%

Spectral–Spatial Hyperspectral Image Classification Using $\ell_{1/2}$ Regularized Low-Rank Representation and Sparse Representation-Based Graph Cuts

Jia

Zhang

2015

IEEE J. Sel. Top. Appl. Earth Observations Remote Sensing

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Hundreds of narrow contiguous spectral bands collected by a hyperspectral sensor have provided the opportunity to identify the various materials present on the surface. Moreover, spatial information, enforcing the assumption that the adjacent pixels belong to the same class with a high probability, is a valuable complement to the spectral information. In this paper, two predominant approaches have been developed to exploit the spatial information. First, by decomposing each pixel and the spatial neighborhood into a low-rank form, the spatial information can be efficiently integrated into the spectral signatures. Meanwhile, in order to describe the low-rank structure of the decomposed data more precisely, an 1/2 norm regularization is introduced and a discrete algorithm is proposed to solve the combined optimization problem by the augmented Lagrange multiplier (ALM) and a halfthreshold operator. Second, a graph cuts segmentation algorithm has been applied on the sparse-representation-based probability estimates of the hyperspectral data to further improve the spatial homogeneity of the material distribution. Experimental results on four real hyperspectral data with different spectral and spatial resolutions have demonstrated the effectiveness and versatility of the proposed spatial information-fused approaches for hyperspectral image classification.Index Terms-Augmented Lagrange multiplier (ALM), hyperspectral image classification, low-rank representation (LRR), nuclear norm.

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$$S_{1/2}$$ S 1 / 2 regularization methods and fixed point algorithms for affine rank minimization problems

Cited by 22 publications

References 36 publications

Sparse Principal Component Analysis via Fractional Function Regularity

Sparse Principal Component Analysis via Fractional Function Regularity

Robust Euclidean embedding via EDM optimization

Spectral–Spatial Hyperspectral Image Classification Using $\ell_{1/2}$ Regularized Low-Rank Representation and Sparse Representation-Based Graph Cuts

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