MM Algorithms for Minimizing Nonsmoothly Penalized Objective Functions

Details MM Algorithms for Minimizing Nonsmoothly Penalized Objective Functions MM Algorithms for Minimizing Nonsmoothly Penalized Objective Functions

2010-01-26

arxiv.org/abs/1001.4776v2

Electronic Journal of Statistics, Volume 4 (2010), pages 1258-1299

Statistics

Computation

A revised version of this paper has been published in the Electronic Journal of Statistics

Scientific paper

10.1214/10-EJS582

In this paper, we propose a general class of algorithms for optimizing an extensive variety of nonsmoothly penalized objective functions that satisfy certain regularity conditions. The proposed framework utilizes the majorization-minimization (MM) algorithm as its core optimization engine. The resulting algorithms rely on iterated soft-thresholding, implemented componentwise, allowing for fast, stable updating that avoids the need for any high-dimensional matrix inversion. We establish a local convergence theory for this class of algorithms under weaker assumptions than previously considered in the statistical literature. We also demonstrate the exceptional effectiveness of new acceleration methods, originally proposed for the EM algorithm, in this class of problems. Simulation results and a microarray data example are provided to demonstrate the algorithm's capabilities and versatility.

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