# A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms
## 1 Introduction  
Currently, there is an increasing interest in developing methods and algorithms that exploit tensor decompositions and modelling [1, 2]. These techniques become of significant importance in many real-world scenarios, e.g., when dealing with large amounts of data, processing multidimensional signals, or solving high-dimensional system identification problems. Many important applications rely on such tensor-based techniques, which can be successfully used in the fields of big data [3], source separation [4], machine learning [5], multiple-input multiple-output (MIMO) communication systems [6], and beamforming [7]. Tensor decompositions and their related applications are frequently addressed based on multilinear signal processing techniques [8, 9]. For example, in the context of system identification scenarios, the problems can be formulated in terms of identifying multilinear forms. As particular cases, we can mention the bilinear and trilinear forms, where the decomposition is performed using two and three components, respectively. Since the Wiener filter and adaptive algorithms represent popular methods to address system identification problems, their applicability was also extended to the multilinear framework. Among the recent related works, we can mention the iterative Wiener filter for bilinear forms [10] and the subsequent adaptive filtering methods [11, 12, 13], together with their extensions to trilinear forms [14, 15, 16, 17]. In this context, the work in [17] provided a system identification framework based on tensor decomposition, which was suitable for the trilinear approach. This work presents the iterative Wiener filter and least-mean-squares (LMS) adaptive algorithms tailored for trilinear forms. Among those,the normalized LMS (NLMS) algorithm for trilinear forms, namely NLMS-TF, represents a practical solution in terms of both performance and complexity. Nevertheless, it is known that the recursive least-squares (RLS) algorithm [18] could provide improved performance as compared to the LMS-based algorithms, especially in terms of the convergence rate. This represents the motivation behind the solution proposed in this paper, which targets the development of the RLS algorithm for trilinear forms, namely RLS-TF. Therefore, the goal of this work is mainly twofold. First, we aim to design a faster converging algorithm as compared to the recently developed NLMS-TF. Second, we intend to outline the performance features of the RLS-TF algorithm as compared to the conventional RLS benchmark. Besides the conventional adaptive algorithms (which usually act as supervised methods), we should also note that unsupervised algorithms can be used in conjunction with tensor-based approaches, e.g., [19; 20]. In this context, we can mention the single-channel blind source separation problem, which targets the identification of individual source signals from a single mixture recording. Such an approach can be found in [19], where the parameters of a so-called \"imitated-stereo\" mixture model (i.e., one real and the other virtual microphones, which results in an artificial mixing system of dual channels) were found by applying tensor estimation and exploiting sparsity features. The method proposed in [20] also exploits sparsity (using the Gibbs distribution) for multichannel source separation, by solving the underdetermined convolutive mixture separation, while considering the reverberations of the surrounding environment. Another appealing field where such tensor-based techniques can be applied is video processing. For example, unsupervised algorithms can be used for detecting anomalies in a video sequence, e.g., detecting micro defects while employing a thermography imaging system [21]. Currently, in many real-world applications related to MIMO systems, speech processing, and image/video processing, the received signals are tensors or can be grouped in a tensorial form. Consequently, as outlined before, utilizing estimation techniques from tensor algebra is beneficial.