A multidimensional shrinkage-thresholding operator

Arnau Tibau Puig, Ami Wiesel, Alfred O. Hero

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

26 Scopus citations


The scalar shrinkage-thresholding operator (SSTO) is a key ingredient of many modern statistical signal processing algorithms including: sparse inverse problem solutions, wavelet denoising, and JPEG2000 image compression. In these applications, it is customary to select the threshold of the operator by solving a scalar sparsity penalized quadratic optimization. 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 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 standard SSTO in the special case of N = 1. In the general case of N > 1 the optimal MSTO threshold can be found by a simple convex line search. We present three illustrative applications of the MSTO in the context of non-linear regression: l 2-penalized linear regression, Group LASSO linear regression and Group LASSO logistic regression.

Original languageAmerican English
Title of host publication2009 IEEE/SP 15th Workshop on Statistical Signal Processing, SSP '09
Number of pages4
StatePublished - 2009
Externally publishedYes
Event2009 IEEE/SP 15th Workshop on Statistical Signal Processing, SSP '09 - Cardiff, United Kingdom
Duration: 31 Aug 20093 Sep 2009

Publication series

NameIEEE Workshop on Statistical Signal Processing Proceedings


Conference2009 IEEE/SP 15th Workshop on Statistical Signal Processing, SSP '09
Country/TerritoryUnited Kingdom


  • Group LASSO regression
  • Iterative group shrinkage-thresholding
  • Multidimensional shrinkage-thresholding operator


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