Multidimensional shrinkage-thresholding operator and group LASSO penalties

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.