Two-Part Reconstruction with Noisy-Sudocodes

We develop a two-part reconstruction framework for signal recovery in compressed sensing (CS), where a fast algorithm is applied to provide partial recovery in Part 1, and a CS algorithm is applied to complete the residual problem in Part 2. Partitioning the reconstruction process into two complementary parts provides a natural trade-off between runtime and reconstruction quality. To exploit the advantages of the two-part framework, we propose a Noisy-Sudocodes algorithm that performs two-part reconstruction of sparse signals in the presence of measurement noise. Specifically, we design a fast algorithm for Part 1 of Noisy-Sudocodes that identifies the zero coefficients of the input signal from its noisy measurements. Many existing CS algorithms could be applied to Part 2, and we investigate approximate message passing (AMP) and binary iterative hard thresholding (BIHT). For Noisy-Sudocodes with AMP in Part 2, we provide a theoretical analysis that characterizes the trade-off between runtime and reconstruction quality. In a 1-bit CS setting where a new 1-bit quantizer is constructed for Part 1 and BIHT is applied to Part 2, numerical results show that the Noisy-Sudocodes algorithm improves over BIHT in both runtime and reconstruction quality.