Fast random vector transforms in terms of pseudo-inverse within the Wiener filtering paradigm

We propose two techniques for the fast numerical implementation of a random vector transform within Wiener filtering paradigm. In signal processing terminology, the transform is interpreted as an optimal filter. The proposed transform aims to minimize the sum of distances between a reference random vector x_i and its transformed counterpart y_i, for all i=1,…,p. In contrast to known techniques where the transform consists of p matrices that minimize a difference between a single reference random vector and p observed random vectors, the proposed transform uses a single optimal matrix to transform p random vectors. It requires less computational work for the numerical implementation. Nevertheless, a direct numerical realization of the transform involves computation of the covariance matrix pseudo-inverse and the matrix square root which might be large, and then are computationally expensive. Proposed fast versions of the transform allow us to avoid that computation and, as a result, accelerate the numerical implementation. While the fast proposed techniques are approximate versions of the original transform, it is shown that their accuracies are very close to those of the original transform. An error analysis is provided. The effectiveness of the proposed transform is illustrated by numerical experiments in four real-world scenarios.