  We study the performance of a Bayesian statistician who estimates a rank-one
signal corrupted by non-symmetric rotationally invariant noise with a generic
distribution of singular values. As the signal-to-noise ratio and the noise
structure are unknown, a Gaussian setup is incorrectly assumed. We derive the
exact analytic expression for the error of the mismatched Bayes estimator and
also provide the analysis of an approximate message passing (AMP) algorithm.
The first result exploits the asymptotic behavior of spherical integrals for
rectangular matrices and of low-rank matrix perturbations; the second one
relies on the design and analysis of an auxiliary AMP. The numerical
experiments show that there is a performance gap between the AMP and Bayes
estimators, which is due to the incorrect estimation of the signal norm.
