Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a complicated matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a converging approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a novel estimation-theoretic interpretation of the CG algorithm. The paper discusses this interpretation and convergence issues and presents numerical examples.
['Michael Schneider', 'Alan S. Willsky']
A Krylov subspace method for large estimation problems