Robust Estimation of Sparse Signal with Unknown Sparsity Cluster Value

Abstract

In the signal+noise model, we assume that the signal has a more general sparsity structure in the sense that the majority of signal coordinates are equal to some value which is assumed to be unknown, contrary to the classical sparsity context where one knows the sparsity cluster value (typically, zero by default). We apply an empirical Bayes approach (linked to the penalization method) for inference on the signal, possibly sparse in this more general sense. The resulting method is robust in that we do not need to know the sparsity cluster value; in fact, the method extracts as much generalized sparsity as there is in the underlying signal. However, as compared to the case of known sparsity cluster value, the proposed robust method cannot be reduced to thresholding procedure anymore. We propose two new procedures: the empirical Bayes model averaging (EBMA) and empirical Bayes model selection (EBMS) procedures, respectively. The former is procedure realized by an MCMC algorithm based on the partial (mixed) normal–normal conjugacy build in our modeling stage, and the latter is based on a new optimization algorithm of $O(n^2)$-complexity. We perform simulations to demonstrate how the proposed procedures work and accommodate possible systematic error in the sparsity cluster value.