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

## TL;DR

This paper extends the theoretical understanding of approximate message passing algorithms by providing performance guarantees for AMP with side information, supported by numerical evidence showing accurate state evolution predictions.

## Contribution

It offers the first rigorous analysis of AMP with side information, establishing conditions under which its performance can be accurately predicted by state evolution.

## Key findings

- Performance guarantees for AMP-SI under joint distribution assumptions
- State evolution accurately predicts mean square error in AMP-SI
- Numerical results support theoretical predictions

## 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 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. An attractive feature of AMP is that when the elements of the signal are exchangeable, the entries of the measurement matrix are independent and identically distributed (i.i.d.) Gaussian, and the denoiser applies the same non-linearity at each entry, the performance of AMP can be predicted accurately by a scalar iteration referred to as state evolution (SE). However, the AMP-SI framework uses different entry-wise scalar denoisers, based on the entry-wise level of the SI, and therefore is not supported by the standard AMP theory. In this work, we provide rigorous performance guarantees for AMP-SI when the input signal and SI are drawn i.i.d. according to some joint distribution subject to finite moment constraints. Moreover, we provide numerical examples to support the theory which demonstrate empirically that the SE can predict the AMP-SI mean square error accurately.