Structured Sparsity Promoting Functions

Lixin Shen, Bruce W. Suter, Erin E. Tripp

Research output: Contribution to journalArticle

Abstract

Motivated by the minimax concave penalty-based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we provide sparsity guarantees for the general family of functions. We further study the behavior of the proximity operators of several special functions, including indicator functions of closed and convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases.

Original languageEnglish (US)
Pages (from-to)386-421
Number of pages36
JournalJournal of Optimization Theory and Applications
Volume183
Issue number2
DOIs
StatePublished - Nov 1 2019

Fingerprint

Sparsity
Moreau Envelope
Indicator function
Special Functions
Variable Selection
Quadratic Function
Closed set
Linear regression
Minimax
Convex Sets
Proximity
Penalty
Linear Combination
High-dimensional
Operator
Mathematical operators
Demonstrate
Guarantee
Variable selection

Keywords

  • Moreau envelope
  • Proximity operator
  • Sparsity
  • Thresholding operator
  • Variable selection

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

Cite this

Structured Sparsity Promoting Functions. / Shen, Lixin; Suter, Bruce W.; Tripp, Erin E.

In: Journal of Optimization Theory and Applications, Vol. 183, No. 2, 01.11.2019, p. 386-421.

Research output: Contribution to journalArticle

Shen, Lixin ; Suter, Bruce W. ; Tripp, Erin E. / Structured Sparsity Promoting Functions. In: Journal of Optimization Theory and Applications. 2019 ; Vol. 183, No. 2. pp. 386-421.
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