State Evolution For Approximate Message Passing With Non-Separable Functions

Given a high-dimensional data matrix A epsilon R-mxn, approximate message passing (AMP) algorithms construct sequences of vectors u(t) epsilon R-n, v(t) epsilon R-m, indexed by t epsilon {0, 1, 2 . . .} by iteratively applying Lambda or Lambda(T) and suitable nonlinear functions, which depend on the specific application. Special instances of this approach have been developed-among other applications-for compressed sensing reconstruction, robust regression, Bayesian estimation, low-rank matrix recovery, phase retrieval and community detection in graphs. For certain classes of random matrices A, AMP admits an asymptotically exact description in the high-dimensional limit m, n -> infinity, which goes under the name of state evolution. Earlier work established state evolution for separable nonlinearities (under certain regularity conditions). Nevertheless, empirical work demonstrated several important applications that require non-separable functions. In this paper we generalize state evolution to Lipschitz continuous non-separable nonlinearities, for Gaussian matrices A. Our proof makes use of Bolthausen's conditioning technique along with several approximation arguments. In particular, we introduce a modified algorithm (called LoAMP for Long AMP), which is of independent interest.