Lower bound theory of nonzero entries in solutions of l2-l p minimization

Xiaojun Chen, Fengmin Xu, Yinyu Ye

Research output: Journal article publicationJournal articleAcademic researchpeer-review

171 Citations (Scopus)

Abstract

Recently, variable selection and sparse reconstruction are solved by finding an optimal solution of a minimization model, where the objective function is the sum of a data-fitting term in l2 norm and a regularization term in lp norm (0 < p < 1). Since it is a nonconvex model, most algorithms for solving the problem can provide only an approximate local optimal solution, where nonzero entries in the solution cannot be identified theoretically. In this paper, we establish lower bounds for the absolute value of nonzero entries in every local optimal solution of the model, which can be used to indentify zero entries precisely in any numerical solution. Therefore, we have developed a lower bound theorem to classify zero and nonzero entries in every local solution. These lower bounds clearly show the relationship between the sparsity of the solution and the choice of the regularization parameter and norm so that our theorem can be used for selecting desired model parameters and norms. Furthermore, we also develop error bounds for verifying the accuracy of numerical solutions of the l2-l p minimization model. To demonstrate applications of our theory, we propose a hybrid orthogonal matching pursuit-smoothing gradient (OMP-SG) method for solving the nonconvex, non- Lipschitz continuous l2-lp minimization problem. Computational results show the effectiveness of the lower bounds for identifying nonzero entries in numerical solutions and the OMP-SG method for finding a high quality numerical solution.
Original languageEnglish
Pages (from-to)2832-2852
Number of pages21
JournalSIAM Journal on Scientific Computing
Volume32
Issue number5
DOIs
Publication statusPublished - 15 Nov 2010

Keywords

  • First order condition
  • L regularization p
  • Linear least-squares problem
  • Second order condition
  • Smoothing approximation
  • Sparse solution
  • Variable selection

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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