An Analysis of State Evolution for Approximate Message Passing with Side Information

Abstract

A common goal in many research areas is to reconstruct an unknown signal x from noisy linear measurements. Approximate message passing (AMP) is a class of low-complexity algorithms that can be used for efficiently solving such high-dimensional regression tasks. Often, it is the case that side information (SI) is available during reconstruction. For this reason, a novel algorithmic framework that incorporates SI into AMP, referred to as approximate message passing with side information (AMP-SI), has been recently introduced. In this work, we provide rigorous performance guarantees for AMP-SI when there are statistical dependencies between the signal and SI pairs and the entries of the measurement matrix are independent and identically distributed (i.i.d.) Gaussian. The AMP-SI performance is shown to be provably tracked by a scalar iteration referred to as state evolution (SE). Moreover, we provide numerical examples that demonstrate empirically that the SE can predict the AMP-SI mean square error accurately. * A subset of this work has been presented at

Community Detection with Contextual Multilayer Networks

“…One possible way to handle it is to treat T as main information and B as side information, or vice versa. Although AMP with side information has been considered in [21] in the context of signal recovery from noisy observations, the generic approach in [21] does not work in the present context. Their side information is separable in the sense that if tS 1 , ¨¨¨, S n u is the set of side information with S i being the side information for node i, then S i 's are mutually independent.…”

“…It is easy to check that b i 's are not independent whenever µ ą 0, and hence the side information is non-separable. This substantially complicates the situation and a direct application of results in [5] as was done in [21] becomes impossible.…”