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.

翻译：我们考虑线性估计问题，并建立了高斯-马尔可夫定理的一种扩展形式，其中允许偏差算子非零，但受限于Schatten类型的矩阵范数。针对核范数和谱范数情形（Frobenius范数情形回归岭回归），我们推导出最优估计量的简洁显式公式。此外，我们解析地推导了多个随机矩阵系综的泛化误差，并与岭回归进行比较。最后，我们进行了大量模拟研究，表明交叉验证的核范数和谱范数回归器在多种情况下可优于岭回归。