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We consider the problem of linear estimation, and establish an extension ofthe Gauss-Markov theorem, in which the bias operator is allowed to be non-zerobut bounded with respect to a matrix norm of Schatten type. We derive simpleand explicit formulas for the optimal estimator in the cases of Nuclear andSpectral norms (with the Frobenius case recovering ridge regression).Additionally, we analytically derive the generalization error in multiplerandom matrix ensembles, and compare with Ridge regression. Finally, we conductan extensive simulation study, in which we show that the cross-validatedNuclear and Spectral regressors can outperform Ridge in several circumstances.

Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem