Mismatched Estimation of rank-one symmetric matrices under Gaussian noise

We consider the estimation of an n-dimensional vector s from the noisy\nelement-wise measurements of $\\mathbf{s}\\mathbf{s}^T$, a generic problem that\narises in statistics and machine learning. We study a mismatched Bayesian\ninference setting, where some of the parameters are not known to the\nstatistician. We derive the full exact analytic expression of the asymptotic\nmean squared error (MSE) in the large system size limit for the particular case\nof Gaussian priors and additive noise. From our formulas, we see that\nestimation is still possible in the mismatched case; and also that the minimum\nMSE (MMSE) can be achieved if the statistician chooses suitable parameters. Our\ntechnique relies on the asymptotics of the spherical integrals and can be\napplied as long as the statistician chooses a rotationally invariant prior.\n