A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms

Abstract

High-dimensional system identification problems can be efficiently addressed based on tensor decompositions and modelling. In this paper, we design a recursive least-squares (RLS) algorithm tailored for the identification of trilinear forms, namely RLS-TF. In our framework, the trilinear form is related to the decomposition of a third-order tensor (of rank one). The proposed RLS-TF algorithm acts on the individual components of the global impulse response, thus being efficient in terms of both performance and complexity. Simulation results indicate that the proposed solution outperforms the conventional RLS algorithm (which handles only the global impulse response), but also the previously developed trilinear counterparts based on the least-mean- squares algorithm.

Tensor-Based Adaptive Filtering Algorithms

“…where 0 < α ≤ 1 denotes the normalized step-size parameter, δ > 0 is the regularization constant, and e(n) was defined in (25). Nevertheless, similar to the previous discussion related to the LMS-T and LMS algorithms, the NLMS-T uses N adaptive filters of lengths L i , i = 1, 2, .…”

“…This kind of decomposition into two such smaller structures (i.e., bilinear forms) has been used before, and the initial system identification problem was addressed using different tools, such as an iterative version of the Wiener filter [21], adaptive algorithms [22,23], and the Kalman filter [24]. Furthermore, the trilinear forms are related to the decomposition of third-order tensors [8,15,25].…”

Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms

“…The tensor-based RLS (RLS-T) algorithm is available in Algorithm 1. This approach has been deeply analyzed in the paper [26] by using N = 3 (i.e., third-order tensors). The current sub-section described a generalization for N ≥ 3, thus providing even more efficient implementation solutions based on RLS adaptive filters.…”

Tensor-Based Recursive Least-Squares Adaptive Algorithms with Low-Complexity and High Robustness Features

“…Despite their prohibitive nature, in classical and tensor-based structures, the RLS methods outperform their popular counterparts, such as the affine projection algorithm (APA) [8] and the algorithms based on the least-mean-square (LMS) method [9][10][11][12][13][14][15], which have been preferred in real-world applications due to their low arithmetic requirements. However, due to the tensor-based approach, the RLS algorithms designed for the identification of bilinear and trilinear forms [16,17] have become computationally efficient, as compared to the conventional LMS solutions. Furthermore, the recently proposed tensor-based RLS algorithm (RLS-T) [5] is tailored for the identification of multilinear forms.…”