  In this paper, we study a spiked Wigner problem with an inhomogeneous noise
profile. Our aim in this problem is to recover the signal passed through an
inhomogeneous low-rank matrix channel. While the information-theoretic
performances are well-known, we focus on the algorithmic problem. We derive an
approximate message-passing algorithm (AMP) for the inhomogeneous problem and
show that its rigorous state evolution coincides with the information-theoretic
optimal Bayes fixed-point equations. We identify in particular the existence of
a statistical-to-computational gap where known algorithms require a
signal-to-noise ratio bigger than the information-theoretic threshold to
perform better than random. Finally, from the adapted AMP iteration we deduce a
simple and efficient spectral method that can be used to recover the transition
for matrices with general variance profiles. This spectral method matches the
conjectured optimal computational phase transition.
