# Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms
## 1 Introduction  
Tensor decomposition is an important topic for the development of modern technologies such as big data, machine learning, communication systems with multiple inputs and/or multiple outputs, respectively source separation/beamforming [1, 2, 3]. When implementing system identification tasks, the problems can be reformulated to split the arithmetic workload into multiple smaller system identification problems. Considerable recent research was dedicated to identify multilinear forms [4, 5, 6] with two, three, or even more components using the Wiener filter [7, 8, 9] and adaptive algorithms based on the least-mean-square (LMS) method [10, 11, 12, 13] or on the recursive least-squares (RLS) systems [14, 15, 10]. Despite the attractive performance of the latter, its classical implementations are considered too costly for most practical applications and the preferred workhorse remains the LMS or the normalized-LMS (NLMS) method, which is not sensitive to the scaling of its input signal.  
The traditional least-squares based adaptive systems require complexities proportional to the second or third power of the filter's length in order to compute the corresponding matrix inverse and the associated coefficients. The filter update employs a full spectrum of mathematical operations, including divisions, and it is often plagued by numerical instability. Despite its prohibitive nature in terms of required chip area, the decorrelation properties of the RLS (translated in practice into superior convergence speeds and overall performance) pushed researchers to develop new corresponding versions with attractive compromise between complexity and performance. However, for some algorithms, such as Fast-RLS (FRLS) [16], the negative aspects still tend to overtake the value brought by the reduction in terms of complexity and deny the practical applications a reliable least-squares algorithm.  
For the system identification implementations, which can be decomposed using multiple-input/single-output (MISO) setups, several tensor based models were recently proposed [11]. One such decomposition uses the RLS method based on Woodburry'sidentity [15] to split the unknown system determination into multiple smaller adaptive systems. Despite the fact that the overall complexity is reduced, the algorithm previously introduced as tensor-based RLS (RLS-T) is still dependent on the square of each filter's length, and it is also prone to inherit the problems of classical least-squares solutions [14].  
Recently, amongst the newly proposed RLS methods, the combination with the dichotomous coordinate descent (DCD) iterations has been established as a possible stable alternative with lower complexity traits, proportional to the adaptive filter's length. This paper proposes a tensorial decomposition for multilinear forms based on the RLS-DCD method, which was extensively studied and widely accepted as an efficient solution for least-squares adaptive systems [17; 18]. The DCD portion of the algorithm exploits the nature of the corresponding correlation matrices and estimates with minimal computational effort, in an iterative manner, an incremental vector for each of the adaptive filter to be updated. The operations are performed using only additions and bit-shifts, and any overall requirement for divisions is eliminated [19; 20]. For the identification of an unknown MISO system based on its tensorial form, multiple RLS-DCD smaller adaptive filters will be employed, which inherit the performance of the classical RLS versions and require lower arithmetic efforts.  
The paper is organized as follows: Section 2 introduces the framework of the multilinear forms. In Section 3, the RLS-T is described with its corresponding notations and the newly RLS-DCD-T adaptive systems are introduced as an alternative for the former. The new adaptive model is validated in Section 4 through several simulation results and conclusions are presented in Section 5.