Subspace Expansion and the Equivalence of Conjugate Direction and Multistage Wiener Filters

We consider iterative subspace Wiener filters for solving minimum mean-squared error (MMSE) and minimum variance unbiased estimation problems in low-dimensional subspaces. In this class of subspace filters, the conjugate direction and multistage Wiener filters comprise two large subclasses, and within these the conjugate gradient and orthogonal multistage Wiener filters are the most prominent. We establish very general equivalences between conjugate direction and multistage Wiener filters, wherein the direction vectors of a conjugate direction filter and the stagewise vectors of a multistage filter are related through a one-term autoregressive recursion. By virtue of this recursion, the expanding subspaces of the two filters are identical, even though their bases for them are different. As a consequence, their subspace filters, gradients, and MSEs are identical at each stage of the subspace iteration. If the conjugate direction filter is a conjugate gradient filter, then the equivalent stagewise filter is an orthogonal multistage filter, and vice-versa. If either the conjugate gradient filter or the orthogonal multistage filter is initialized at the cross-covariance vector between the signal and the measurement, then each of the subspace filters iteratively turns out a basis for a Krylov subspace.