Binary vector reconstruction via discreteness-aware approximate message passing

Abstract

In this paper, we propose an algorithm to reconstruct a binary vector from its possibly underdetermined linear measurements. Taking advantage of the idea of the approximate message passing (AMP) algorithm for compressed sensing, we firstly formulate a probability distribution associated with the sum-of-absolute-value (SOAV) optimization. Then, by approximating the sum-product algorithm for the marginalization of the distribution, we obtain a low-complexity iterative algorithm, called discreteness-aware AMP (DAMP). We evaluate the performance of DAMP analytically via state evolution and derive a condition for the exact reconstruction with DAMP. Moreover, we also provide Bayes optimal DAMP for the binary vector reconstruction, which gives the minimum mean-square-error at each iteration in the large system limit. Simulation results show that DAMP can reconstruct the binary vector from underdetermined linear measurements and its performance can be well predicted by our theoretical results.