Multidimensional shrinkage-thresholding operator and group LASSO penalties

Arnau Tibau Puig, Ami Wiesel, Gilles Fleury, Alfred O. Hero

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


The scalar shrinkage-thresholding operator is a key ingredient in variable selection algorithms arising in wavelet denoising, JPEG2000 image compression and predictive analysis of gene microarray data. In these applications, the decision to select a scalar variable is given as the solution to a scalar sparsity penalized quadratic optimization. In some other applications, one seeks to select multidimensional variables. In this work, we present a natural multidimensional extension of the scalar shrinkage thresholding operator. Similarly to the scalar case, the threshold is determined by the minimization of a convex quadratic form plus an Euclidean norm penalty, however, here the optimization is performed over a domain of dimension N ≥ 1. The solution to this convex optimization problem is called the multidimensional shrinkage threshold operator (MSTO). The MSTO reduces to the scalar case in the special case of N=1. In the general case of N > 1 the optimal MSTO shrinkage can be found through a simple convex line search. We give an efficient algorithm for solving this line search and show that our method to evaluate the MSTO outperforms other state-of-the art optimization approaches. We present several illustrative applications of the MSTO in the context of Group LASSO penalized estimation.

Original languageAmerican English
Article number5742974
Pages (from-to)363-366
Number of pages4
JournalIEEE Signal Processing Letters
Issue number6
StatePublished - 2011

Bibliographical note

Funding Information:
Manuscript received February 14, 2011; accepted March 16, 2011. Date of publication April 07, 2011; date of current version April 21, 2011. The work of A. O. Hero, III, G. Fleury, and A. Tibau Puig was supported in part by the Dig-iteo DANSE project. The work of A. Wiesel was supported by a Marie Curie Outgoing International Fellowship within the 7th European Community Frame-work Programme. This work was presented in part at the IEEE/SP 15th Workshop on Statistical Signal Processing, August 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Hsiao-Chun Wu.


  • Shrinkage-thresholding operator
  • group LASSO regression
  • proximity operator


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