Mismatched Estimation of Non-Symmetric Rank-One Matrices Under Gaussian Noise

We consider the estimation of a nxm matrix uv(T) observed through an additive Gaussian noise channel, a problem that frequently arises in statistics and machine learning. We investigate a scenario involving mismatched Bayesian inference in which the statistician is unaware of true prior and uses an assumed prior. We derive the exact analytic expression for the asymptotic mean squared error (MSE) in the large system size limit for the particular case of Gaussian priors and additive noise. Our formulas demonstrate that in the mismatched case, estimation is still possible. Additionally, the minimum MSE (MMSE) can be obtained by selecting a non-trivial set of parameters beyond the matched parameters. Our technique is based on the asymptotic behavior of spherical integrals for rectangular matrices. Our method can be extended to non-rotation-invariant distributions for the true prior but requires rotation invariance for the statistician's assumed prior.