Robust Simultaneous Sparse Approximation

This chapter considers sparse underdetermined (ill-posed) multivariate multiple linear regression model known in signal processing literature as multiple measurement vector (MMV) model. The objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vectors. The MMV model is an extension of the compressed sensing (CS) which is an emerging field that has attracted considerable research interest over the past few years. Recently, many popular greedy pursuit algorithms have been extended to MMV setting. All these methods, such as simultaneous normalized iterative hard thresholding (SNIHT), are not resistant to outliers or heavy-tailed errors. In this chapter, we develop a robust SNIHT method that computes the estimates of the sparse signal matrix and the scale of the error distribution simultaneously. The method is based on Huber’s criterion and hence referred to as HUB-SNIHT algorithm. The method can be tuned to have a negligible performance loss compared to SNIHT under Gaussian noise, but obtains superior joint sparse recovery under heavy-tailed non-Gaussian noise conditions.