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Structured Sparsity Promoting Functions

Author

Listed:
  • Lixin Shen

    (Syracuse University)

  • Bruce W. Suter

    (Air Force Research Laboratory)

  • Erin E. Tripp

    (Syracuse University)

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.

Suggested Citation

  • Lixin Shen & Bruce W. Suter & Erin E. Tripp, 2019. "Structured Sparsity Promoting Functions," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 386-421, November.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:2:d:10.1007_s10957-019-01565-0
    DOI: 10.1007/s10957-019-01565-0
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. Chayne Planiden & Xianfu Wang, 2018. "Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 21-63, April.
    3. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    4. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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