$NP/CMP$  Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$  Minimization and $l_{p}$  Minimization for Information Processing

Peng, Jigen; Yue, Shigang; Li, Haiyang

doi:10.1109/tit.2015.2429611

Cited by 35 publications

(30 citation statements)

References 28 publications

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“…Theorem 2 is the main theorem in [7]. Obviously, this theorem qualitatively proves the effectiveness of solving the original -minimization problem via -minimization.…”

Section: Preliminariesmentioning

confidence: 69%

“…It is shown that is a continuous function in when , where is the smallest number of columns from A that are linearly dependent. Therefore, if for some fixed A and k , then there exists a constant such that for , that is, every k -sparse vector can be recovered via both -minimization and -minimization for , which is a corollary of the main theorem in [7]. …”

Section: Preliminariesmentioning

confidence: 99%

“…Recently, Peng, Yue, and Li [7] have proved that there exists a constant such that every solution of -minimization is also a solution of -minimization whenever . This result builds a bridge between -minimization and -minimization, and it is important that this conclusion is not limited by the structure of a matrix A .…”

Section: Introductionmentioning

confidence: 98%

“…To find the sparsest one, much excellent theoretical work (see, e.g . [5, 6], and [7]) has been devoted to the -minimization. However, Natarajan [8] proved that -minimization is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

“…This result builds a bridge between -minimization and -minimization, and it is important that this conclusion is not limited by the structure of a matrix A . However, the paper [7] does not give an analytic expression of . The model of choice of -minimization is still difficult.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the equivalence relationship between l 0 $l_{0}$ -minimization and l p

Wang

Peng

2017

J Inequal Appl

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In signal processing theory, -minimization is an important mathematical model. Unfortunately, -minimization is actually NP-hard. The most widely studied approach to this NP-hard problem is based on solving -minimization (). In this paper, we present an analytic expression of , which is formulated by the dimension of the matrix , the eigenvalue of the matrix , and the vector , such that every k-sparse vector can be exactly recovered via -minimization whenever , that is, -minimization is equivalent to -minimization whenever . The superiority of our results is that the analytic expression and each its part can be easily calculated. Finally, we give two examples to confirm the validity of our conclusions.

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“…Theorem 2 is the main theorem in [7]. Obviously, this theorem qualitatively proves the effectiveness of solving the original -minimization problem via -minimization.…”

Section: Preliminariesmentioning

confidence: 69%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…To find the sparsest one, much excellent theoretical work (see, e.g . [5, 6], and [7]) has been devoted to the -minimization. However, Natarajan [8] proved that -minimization is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of the equivalence relationship between l 0 $l_{0}$ -minimization and l p

Wang

Peng

2017

J Inequal Appl

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Compressed Sensing and Dictionary Learning

Swamy

2019

Neural Networks and Statistical Learning

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The sparsest solution of the union of finite polytopes via its nonconvex relaxation

2019

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Sparse optimization problems have gained much attention since 2004. Many approaches have been developed, where nonconvex relaxation methods have been a hot topic in recent years. In this paper, we study a partially sparse optimization problem, which finds a partially sparsest solution of a union of finite polytopes. We discuss the relationship between its solution set and the solution set of its nonconvex relaxation. In details, by using geometrical properties of polytopes and properties of a family of well-defined nonconvex functions, we show that there exists a positive constant p * ∈ (0, 1] such that for every p ∈ [0, p * ), all optimal solutions to the nonconvex relaxation with the parameter p are also optimal solutions to the original sparse optimization problem. This provides a theoretical basis for solving the underlying problem via its nonconvex relaxation. Moreover, we show that the problem we concerned covers a wide range of problems so that several important sparse optimization problems are its subclasses. Finally, by an example we illustrate our theoretical findings.

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$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

Cited by 35 publications

References 28 publications

Analysis of the equivalence relationship between l 0 $l_{0}$ -minimization and l p

Analysis of the equivalence relationship between l 0 $l_{0}$ -minimization and l p

Compressed Sensing and Dictionary Learning

The sparsest solution of the union of finite polytopes via its nonconvex relaxation

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