Group sparse optimization via ℓp;qregularization

Yaohua Hu, Chong Li, Kaiwen Meng, Jing Qin, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

71 Citations (Scopus)


In this paper, we investigate a group sparse optimization problem via ℓp;qregularization in three aspects: Theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish an oracle property and a global recovery bound of order O(λ 2 2/2-q ) for any point in a level set of the ℓp;qregularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order O(λ2) for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the ℓp;qregularization problems, either by analytically solving some specific p;q regularization subproblems, or by using the Newton method to solve general ℓp;qregularization subproblems. In particular, we establish a local linear convergence rate of the proximal gradient method for solving the ℓp;qregularization problem under some mild conditions and by first proving a second-order growth condition. As a consequence, the local linear convergence rate of proximal gradient method for solving the usual ℓl;qregularization problem (0 < q < 1) is obtained. Finally in the aspect of application, we present some numerical results on both the simulated data and the real data in gene transcriptional regulation.
Original languageEnglish
JournalJournal of Machine Learning Research
Publication statusPublished - 1 Apr 2017


  • Gene regulation network
  • Group sparse optimization
  • Iterative thresholding algorithm
  • Lower-order regularization
  • Nonconvex optimization
  • Proximal gradient method
  • Restricted eigenvalue condition

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


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