Convolutional Approximate Message-Passing

This letter proposes a novel message-passing algorithm for signal recovery in compressed sensing. The proposed algorithm solves the disadvantages of approximate message-passing (AMP) and orthogonal/vector AMP, and realizes their advantages. AMP converges only in a limited class of sensing matrices while it has low complexity. Orthogonal/vector AMP requires a high-complexity matrix inversion while it is applicable for a wide class of sensing matrices. The key feature of the proposed algorithm is the so-called Onsager correction via a convolution of messages in all preceding iterations while the conventional message-passing algorithms have correction terms that depend only on messages in the latest iteration. Thus, the proposed algorithm is called convolutional AMP (CAMP). Ill-conditioned sensing matrices are simulated as an example in which the convergence of AMP is not guaranteed. Numerical simulations show that CAMP can improve the convergence property of AMP and achieve high performance comparable to orthogonal/vector AMP in spite of low complexity comparable to AMP.