Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem
This work addresses linear estimation methods for statistical modeling, offering incremental improvements by extending classical results to new norm constraints.
The authors tackled the problem of linear estimation by extending the Gauss-Markov theorem to allow bounded bias operators, deriving optimal estimators for Nuclear and Spectral norms and analyzing generalization error in random matrix ensembles. They showed through simulations that cross-validated Nuclear and Spectral regressors can outperform Ridge regression in some cases, with specific performance gains demonstrated in the study.
We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances.