Rigorous State Evolution Analysis for Approximate Message Passing with Side Information
This work offers theoretical validation for a low-complexity algorithm in high-dimensional signal reconstruction with side information, which is incremental as it extends existing AMP analysis.
The paper provides rigorous performance guarantees for approximate message passing with side information (AMP-SI), showing that its mean square error can be accurately predicted by a scalar state evolution iteration under Gaussian measurement matrices and statistical dependencies between signal and side information.
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 Gaussian. The AMP-SI performance is shown to be provably tracked by a scalar iteration referred to as state evolution. Moreover, we provide numerical examples that demonstrate empirically that the SE can predict the AMP-SI mean square error accurately.