Estimating Rank-One Matrices with Mismatched Prior and Noise: Universality and Large Deviations

We prove a universality result that reduces the free energy of rank-one matrix estimation problems in the setting of mismatched prior and noise to the computation of the free energy for a modified Sherrington–Kirkpatrick spin glass. Our main result is an almost sure large deviation principle for the overlaps between the true signal and the estimator for both the Bayes-optimal and mismatched settings. Through the large deviations principle, we recover the limit of the free energy in mismatched inference problems and the universality of the overlaps.