Mismatched Estimation of rank-one symmetric matrices under Gaussian noise
This work addresses a fundamental estimation problem in statistics and machine learning, but it is incremental as it extends existing analysis to mismatched cases with specific assumptions.
The paper tackles the problem of estimating a rank-one symmetric matrix from noisy measurements in a mismatched Bayesian setting, deriving exact asymptotic mean squared error formulas for Gaussian priors and additive noise, showing that estimation remains possible and the minimum MSE can be achieved with suitable parameter choices.
We consider the estimation of an n-dimensional vector s from the noisy element-wise measurements of $\mathbf{s}\mathbf{s}^T$, a generic problem that arises in statistics and machine learning. We study a mismatched Bayesian inference setting, where some of the parameters are not known to the statistician. We derive the full exact analytic expression of the asymptotic mean squared error (MSE) in the large system size limit for the particular case of Gaussian priors and additive noise. From our formulas, we see that estimation is still possible in the mismatched case; and also that the minimum MSE (MMSE) can be achieved if the statistician chooses suitable parameters. Our technique relies on the asymptotics of the spherical integrals and can be applied as long as the statistician chooses a rotationally invariant prior.