Nonlinear adaptive algorithms with tensor rank decompositions. - Felipe Chaud Pinheiro (2021)
In this work, we develop a theory of adaptive filters whose filtering structure and the corresponding input present some multilinear and/or tensorial relation in their coefficients. Such structures are highly nonlinear, which turns the analysis quite challenging. Nevertheless, we develop techniques that allow for studying a wide class of these algorithms. The work adopts a generic formulation using a new concept, the multitensors, which are, simply put, a sum of tensors of different orders. The system under study has its output defined as the contraction of the input multitensor and the parameter multitensor. Different restrictions imposed on the input and/or on the parameter multitensors result in a myriad of different models and corresponding adaptive algorithms that are analyzed in details, unveiling computational complexity reductions (expressive in some cases), convergence performance and stability, steady-state error, efficient implementation techniques and competitive advantages. Several important works from the literature are generalized and unified under our multitensorial formulation, achieving a wide range of applications. This study presents a review of concepts from multilinear algebra and tensors, which allows us to define all the classes of systems that will be considered here. In the sequel, such systems are studied in the context of Estimation Theory. Some exact gradientdescent methods are developed to find solutions for the nonlinear estimation problems previously defined for all classes covered in this work. They are: the gradient-descent method, the Newtons method and a normalized version of the gradient-descent method. After that, classical approximations for the signals statistics leads to the stochastic gradient algorithms counterpartsthe adaptive filters. In particular, the algorithms are: the least-mean squares (LMS), the SLMS (Stabilized LMS), the normalized LMS (NLMS), the affine projections (APA), the Ture-LMS (An LMS variant with multiple input data) and the stabilized True-LMS. Theoretical analysis for the mean-square error (MSE) are obtained and compared to simulations. Comparisons to several well known algorithms from the literature are also presented, showing advantages for the methods developed here. A certain fluency in linear and abstract algebras are assumed, although the main concepts are introduced in the text and in the appendices.