# A Recursive Least-Squares with a Time-Varying Regularization Parameter
## 1 Introduction  
Least-squares (LS) estimation methods emerge crucially in many applied sciences and engineering applications such as geodesy, adaptive filters, statistics, signal processing and control theory, to name a few. It is widespread due to its simple and efficient implementation. The standard LS problem dates back to 1795 AD when [PERSON] used it to study planetary motions [1]. An obvious advantage of this estimation method is that it does not require prior information on data; only a signal model is assumed [2]. In many applications, data are increased progressively in time. In this case, data can be processed for least-squares problems in two manners, either waiting for all the available data to be collected or processing them sequentially in time. The latter is known as recursive least-squares (RLS), or sequential least squares [2]. The RLS algorithm is one of the most significant advancements of the LS estimation in the twentieth century, which [PERSON] also conceived in his work on celestial bodies. The original work on the RLS algorithm in modern times is often credited to [PERSON] [3]. When working on LS problems, there are some cases where regularization is needed to compromise solution biasing and algorithmic robustness [4]. Different regularized LS and regularized RLS algorithms have emerged. One of these algorithms is the standard regularized exponentially weighted RLS that employs a regularization matrix whose elements fade exponentially to zero. However, some applications require maintaining regularization throughout the adaptive estimation process, such as beamforming. In addition, regularization ensures the existence of the sample covariance matrix inverse when the matrix has fewer rows than columns or is rank deficient. Regularization has a rich history in matrix computation and numerical analysis methods [5]. Tikhonov regularization methods are among the famous and earliest references on the regularization topic [6]. The literature concerning regularized RLS is quite limited because of the complexity associated with implementation [7]. In many research works, the regularization term does not fade but is treated fixed in all recursion steps, e.g., [8; 9]. However, in some cases, such as adaptive beamforming, this regularization term needs to be adjusted, i.e., time-varying, to cope with changes in the data or noise [10; 11]. The work in [12] proposes a special regularization structure that is updated in time using rank-1 matrices. The structure allows inflation or deflation of the regularization matrix and can ensure, on average, the invertibility of the sample covariance matrix with reasonable complexity. However, the sample covariance matrix can be positive-semidefinite, i.e., its smallest eigenvalue can become zero. The work presented in [4] proposes a time-varying regularized RLS with a generic structure. However, its implementation is complex. Different variations of the regularized least squares for many applications are presented in the literature. For acoustic echo cancellation, the problems of acoustic impulse response estimation and nonlinearity modeling are addressed in [13] with a regularized RLS solution to handle these problems. A variable regularization technique is presented in [14] for the fast affine projection algorithm. The work in [15] introduced an algorithm that accounts for a cost function with a time-varying weighted regularization RLS using a sliding window approach. Another parameter that is related to the regularization parameter is the forgetting factor parameter. In some cases, it is preferable to keep the newer data and ignore the older ones by using a forgetting factor parameter which applies higher weighting to recent data [16]. The problem of updating the forgetting factor parameter recursively in time can be found, for example, in [17; 18; 19]. In this paper, we consider the problem of solving regularized least-squares recursively. The recursion is viewed as a time-update process, which means that the number of observations increases as time progresses.