Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem
Simon Segert
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.