Average Performance Analysis for Thresholding

TL;DR: It is shown that with high probability, the thresholding algorithm can recover signals that are sparse in a redundant dictionary as long as the 2-Babel function is growing slowly, implying that it can succeed for sparsity levels up to the order of the ambient dimension.

Abstract: In this letter, we show that with high probability, the thresholding algorithm can recover signals that are sparse in a redundant dictionary as long as the 2-Babel function is growing slowly. This implies that it can succeed for sparsity levels up to the order of the ambient dimension. The theoretical bounds are illustrated with numerical simulations. As an application of the theory, sensing dictionaries for optimal average performance are characterized, and their performance is tested numerically.