Algorithm unfolding for block-sparse and MMV problems with reduced training overhead

Abstract

In this study, we consider algorithm unfolding for the multiple measurement vector (MMV) problem in the case where only few training samples are available. Algorithm unfolding has been shown to empirically speed-up in a data-driven way the convergence of various classical iterative algorithms, but for supervised learning, it is important to achieve this with minimal training data. For this, we consider learned block iterative shrinkage thresholding algorithm (LBISTA) under different training strategies. To approach almost data-free optimization at minimal training overhead, the number of trainable parameters for algorithm unfolding has to be substantially reduced. We therefore explicitly propose a reduced-size network architecture based on the Kronecker structure imposed by the MMV observation model and present the corresponding theory in this context. To ensure proper generalization, we then extend the analytic weight approach by Liu and Chen to LBISTA and the MMV setting. Rigorous theoretical guarantees and convergence results are stated for this case. We show that the network weights can be computed by solving an explicit equation at the reduced MMV dimensions which also admits a closed-form solution. Toward more practical problems, we then considered convolutional observation models and show that the proposed architecture and the analytical weight computation can be further simplified and thus open new directions for convolutional neural networks. Finally, we evaluate the unfolded algorithms in numerical experiments and discuss connections to other sparse recovering algorithms.