A Nonlinear Adaptive Filter Based on the Model of Simple Multilinear Functionals

TL;DR

This paper introduces a simple nonlinear adaptive filtering model based on multilinear functionals, utilizing tensor algebra and Kronecker products, resulting in an LMS-like algorithm with improved convergence and computational efficiency.

Contribution

It proposes a novel nonlinear filter model using tensor algebra, deriving an LMS-like algorithm and analyzing its optimization landscape and complexity.

Findings

Favorable convergence compared to existing polynomial algorithms

Reduced computational complexity

Effective in system identification tasks

Abstract

Nonlinear adaptive filtering allows for modeling of some additional aspects of a general system and usually relies on highly complex algorithms, such as those based on the Volterra series. Through the use of the Kronecker product and some basic facts of tensor algebra, we propose a simple model of nonlinearity, one that can be interpreted as a product of the outputs of K FIR linear filters, and compute its cost function together with its gradient, which allows for some analysis of the optimization problem. We use these results it in a stochastic gradient framework, from which we derive an LMS-like algorithm and investigate the problems of multi-modality in the mean-square error surface and the choice of adequate initial conditions. Its computational complexity is calculated. The new algorithm is tested in a system identification setup and is compared with other polynomial algorithms from the literature, presenting favorable convergence and/or computational complexity.