  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.
