Multilinear Karhunen-Loève Transforms

Abstract

Many recent applications involve distributed signal processing where a source signal is observed by, say, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> local receiver-transmitters and then transmitted to a reconstruction center for the signal estimation. An optimal determination of the receiver-transmitters and the reconstruction center requires extensions of the Karhunen-Loève transform (KLT) and Wiener filter. In this paper, the associated extensions are provided. The proposed optimal multilinear filter is a generalization of the Wiener filter and consists of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> terms where each term is associated with a local receiver-transmitter. For the case when the receiver-transmitters must reduce the dimensionality of the observed signals, two associated techniques are proposed: the multilinear KLT-1 and multilinear KLT-2. The multilinear KLT-1 is constructed in terms of pseudo-inverse matrices and therefore always exists. The multilinear KLT-2 is given in terms of non-singular matrices and it may provide a higher associated accuracy than that of the multilinear KLT-1. All three proposed techniques are based on a reduction of the original problem to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> separate error minimization problems with small matrices. This allows us to provide a fast computational procedure for the multilinear filter, and decrease the computational cost for constructing the multilinear KLT-1 and KLT-2.