MV-PURE Estimator: Minimum-Variance Pseudo-Unbiased Reduced-Rank Estimator for Linearly Constrained Ill-Conditioned Inverse Problems

This paper proposes a novel estimator named minimum-variance pseudo-unbiased reduced-rank estimator (MV- PURE) for the linear regression model, designed specially for the case where the model matrix is ill-conditioned and the unknown deterministic parameter vector to be estimated is subjected to known linear constraints. As a natural generalization of the Gauss-Markov (BLUE) estimator, the MV-PURE estimator is a solution of the following hierarchical nonconvex constrained optimization problem directly related to the mean square error expression. In the first-stage optimization, under a rank constraint, we minimize simultaneously all unitarily invariant norms of an operator applied to the unknown parameter vector in view of suppressing bias of the proposed estimator. Then, in the second-stage optimization, among all pseudo-unbiased reduced-rank estimators defined as the solutions of the first-stage optimization, we find the one achieving minimum variance. We derive a closed algebraic form of the MV-PURE estimator and show that well-known estimators-the Gauss-Markov (BLUE) estimator, the generalized Marquardt´s reduced-rank estimator, and the minimum-variance conditionally unbiased affine estimator subject to linear restrictions-are all special cases of the MV-PURE estimator. We demonstrate the effectiveness of the proposed estimator in a numerical example, where we employ the MV-PURE estimator to the ill-conditioned problem of reconstructing a 2-D image subjected to linear constraints from blurred, noisy observation. This example demonstrates that the MV-PURE estimator outperforms all aforementioned estimators, as it achieves smaller mean square error for all values of signal-to-noise ratio.