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We consider the problem of estimating a rank-1 signal corrupted by structuredrotationally invariant noise, and address the following question: how well doinference algorithms perform when the noise statistics is unknown and henceGaussian noise is assumed? While the matched Bayes-optimal setting withunstructured noise is well understood, the analysis of this mismatched problemis only at its premises. In this paper, we make a step towards understandingthe effect of the strong source of mismatch which is the noise statistics. Ourmain technical contribution is the rigorous analysis of a Bayes estimator andof an approximate message passing (AMP) algorithm, both of which incorrectlyassume a Gaussian setup. The first result exploits the theory of sphericalintegrals and of low-rank matrix perturbations; the idea behind the second oneis to design and analyze an artificial AMP which, by taking advantage of theflexibility in the denoisers, is able to "correct" the mismatch. Armed withthese sharp asymptotic characterizations, we unveil a rich and often unexpectedphenomenology. For example, despite AMP is in principle designed to efficientlycompute the Bayes estimator, the former is outperformed by the latter in termsof mean-square error. We show that this performance gap is due to an incorrectestimation of the signal norm. In fact, when the SNR is large enough, theoverlaps of the AMP and the Bayes estimator coincide, and they even match thoseof optimal estimators taking into account the structure of the noise.

The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?