Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise
This work addresses estimation errors in statistical models with mismatched assumptions, which is an incremental contribution relevant for researchers in signal processing and machine learning dealing with noisy data.
The paper tackles the problem of estimating a rank-one signal corrupted by structured noise when the signal-to-noise ratio and noise structure are unknown, leading to a mismatched Gaussian assumption; it derives exact analytic expressions for the error of the mismatched Bayes estimator and analyzes an approximate message passing algorithm, with numerical experiments showing a performance gap due to incorrect signal norm estimation.
We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.