Tensor Wiener Filter

In signal processing and data analytics, Wiener filter is a classical powerful tool to transform an input signal to match a desired or target signal by a linear time-invariant (LTI) filter. The input signal of a Wiener filter is one-dimensional while its associated least-squares solution, namely Wiener-Hopf equation, involves a two-dimensional data-array, or correlation matrix. However, the actual match should often be carried out between a multi-dimensional filtered signal-sequence, which is the output of a multi-channel filter characterized as a linear-time-invariant MIMO (multi-input and multi-output) system, and a multi-dimensional desired signal-sequence simultaneously. In the presence of such a multi-channel filter, the solution to the corresponding Wiener filter, which we call MIMO Wiener-Hopf equation now, involves a correlation tensor. Therefore, we call this optimal multi-channel filter Tensor Wiener Filter (TWF). Due to lack of the pertinent mathematical framework of needed tensor operations, TWF has never been investigated so far. Now we would like to make the first-ever attempt to establish a new mathematical framework for TWF, which relies on the inverse of the correlation tensor. We propose the new parallel block-Jacobi tensor-inversion algorithm for this tensor inversion. A typical application of the new TWF approach is illustrated as a multi-channel linear predictor (MCLP) built upon a multi-channel autoregressive (MCAR) filter with multi-dimensional input data. Numerical experiments pertaining to seismic data, optical images, and macroeconomic time-series are conducted in comparison with other existing methods. The memory- and computational-complexities corresponding to our proposed parallel block-Jacobi tensor-inversion algorithm are also studied in this paper.