On the Convergence of Convolutional Approximate Message-Passing for Gaussian Signaling

抄録

Convolutional approximate message-passing (CAMP) is an efficient algorithm to solve linear inverse problems. CAMP aims to realize advantages of both approximate message-passing (AMP) and orthogonal/vector AMP. CAMP uses the same low-complexity matched-filter as AMP. To realize the asymptotic Gaussianity of estimation errors for all right-orthogonally invariant matrices, as guaranteed in orthogonal/vector AMP, the Onsager correction in AMP is replaced with a convolution of all preceding messages. CAMP was proved to be asymptotically Bayes-optimal if a state-evolution (SE) recursion converges to a fixed-point (FP) and if the FP is unique. However, no proofs for the convergence were provided. This paper presents a theoretical analysis for the convergence of the SE recursion. Gaussian signaling is assumed to linearize the SE recursion. A condition for the convergence is derived via a necessary and sufficient condition for which the linearized SE recursion has a unique stationary solution. The SE recursion is numerically verified to converge toward the Bayes-optimal solution if and only if the condition is satisfied. CAMP is compared to conjugate gradient (CG) for Gaussian signaling in terms of the convergence properties. CAMP is inferior to CG for matrices with a large condition number while they are comparable to each other for a small condition number. These results imply that CAMP has room for improvement in terms of the convergence properties.