On the convergence of approximate message passing with arbitrary matrices

Approximate message passing (AMP) methods and their variants have attracted considerable recent attention for the problem of estimating a random vector x observed through a linear transform A. In the case of large i.i.d. A, the methods exhibit fast convergence with precise analytic characterizations on the algorithm behavior. However, the convergence of AMP under general transforms is not fully understood. In this paper, we provide sufficient conditions for the convergence of a damped version of the generalized AMP (GAMP) algorithm in the case of Gaussian distributions. It is shown that, with sufficient damping the algorithm can be guaranteed to converge, but the amount of damping grows with peak-to-average ratio of the squared singular values of A. This condition explains the good performance of AMP methods on i.i.d. matrices, but also their difficulties with other classes of transforms. A related sufficient condition is then derived for the local stability of the damped GAMP method under more general (possibly non-Gaussian) distributions, assuming certain strict convexity conditions.