Minimum-Variance Pseudo-Unbiased Low-Rank Estimator for Ill-Conditioned Inverse Problems

This paper presents a mathematically novel low-rank linear statistical estimator named minimum-variance pseudo-unbiased low-rank estimator for applications to ill-conditioned linear inverse problems. Based on a simple fact: `any low-rank estimator can not be a (uniformly) unbiased estimator', we introduce pseudo-unbiased low-rank estimator, as an ideal low-rank extension of unbiased estimators. The minimum-variance pseudo-unbiased low-rank estimator minimizes the variance of estimate among all pseudo-unbiased low-rank estimators, hence it is characterized as a solution to a double layered nonconvex optimization problem. The main theorem presents an algebraic structure of the minimum-variance pseudo-unbiased low-rank estimator in terms of the singular value decomposition of the model matrix in the linear statistical model. The minimum-variance pseudo-unbiased low-rank estimator is not only a best low-rank extension of the minimum-variance unbiased estimator (i.e., Gauss-Markov estimator) but also a nontrivial generalization of the Marquardt's low-rank estimator (Marquardt 1970)