TY - GEN

T1 - A multidimensional shrinkage-thresholding operator

AU - Puig, Arnau Tibau

AU - Wiesel, Ami

AU - Hero, Alfred O.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Group LASSO regression

KW - Iterative group shrinkage-thresholding

KW - Multidimensional shrinkage-thresholding operator

UR - http://www.scopus.com/inward/record.url?scp=72349092082&partnerID=8YFLogxK

U2 - 10.1109/SSP.2009.5278625

DO - 10.1109/SSP.2009.5278625

M3 - Conference contribution

AN - SCOPUS:72349092082

SN - 9781424427109

T3 - IEEE Workshop on Statistical Signal Processing Proceedings

SP - 113

EP - 116

BT - 2009 IEEE/SP 15th Workshop on Statistical Signal Processing, SSP '09

T2 - 2009 IEEE/SP 15th Workshop on Statistical Signal Processing, SSP '09

Y2 - 31 August 2009 through 3 September 2009

ER -