Krylov subspace estimation

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 e ciently using a Krylov subspace method, such as the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is di cult because the error variances are the diagonal elements of a matrix expression involving the inverse of a given matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a convergent approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a new estimation-theoretic interpretation of the CG algorithm. This paper discusses this interpretation and convergence issues and presents numerical examples. The examples include a 10-dimensional estimation problem from oceanography.