  We consider the problem of estimating a rank-1 signal corrupted by structured
rotationally invariant noise, and address the following question: how well do
inference algorithms perform when the noise statistics is unknown and hence
Gaussian noise is assumed? While the matched Bayes-optimal setting with
unstructured noise is well understood, the analysis of this mismatched problem
is only at its premises. In this paper, we make a step towards understanding
the effect of the strong source of mismatch which is the noise statistics. Our
main technical contribution is the rigorous analysis of a Bayes estimator and
of an approximate message passing (AMP) algorithm, both of which incorrectly
assume a Gaussian setup. The first result exploits the theory of spherical
integrals and of low-rank matrix perturbations; the idea behind the second one
is to design and analyze an artificial AMP which, by taking advantage of the
flexibility in the denoisers, is able to "correct" the mismatch. Armed with
these sharp asymptotic characterizations, we unveil a rich and often unexpected
phenomenology. For example, despite AMP is in principle designed to efficiently
compute the Bayes estimator, the former is outperformed by the latter in terms
of mean-square error. We show that this performance gap is due to an incorrect
estimation of the signal norm. In fact, when the SNR is large enough, the
overlaps of the AMP and the Bayes estimator coincide, and they even match those
of optimal estimators taking into account the structure of the noise.
