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

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