Simultaneous Tracking of the Best Basis in Reduced-Rank Wiener Filter

A new on-line learning algorithm that yields a reduced-rank Wiener filter (RRWF) is proposed. The RRWF is defined as the matrix of prescribed rank that provides the best least-squares approximation of a given signal. This implies that an RRWF determines only a subspace, but is not endowed with information of basis functions or axes for the subspace. In other words, even if we want to reduce the rank of the estimated RRWF, we should learn another RRWF of "more reduced rank" again. Our goal in this paper is therefore to establish a learning rule that simultaneously tracks basis functions yielding a matrix that gives an RRWF. To this end, we reformulate the optimization problem of RRWFs, which will be solved by a gradient-based algorithm derived within the framework of differential geometry. Numerical examples are illustrated to support the proposals in the paper