A Multidimensional Shrinkage-Thresholding Operator

Abstract : 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: l2-penalized linear regression, Group LASSO linear regression and Group LASSO logistic regression.
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Arnau Tibau Puig, Ami Wiesel, Alfred O. Hero. A Multidimensional Shrinkage-Thresholding Operator. IEEE Workshop on Statistical Signal Processing (SSP'09), Aug 2009, Cardiff, United Kingdom. pp.113-116. ⟨hal-00444255⟩

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