Fast Sparse RLS Algorithms

Zhen Qin, Jun Tao, Liang An, Shuai Yao, Xiao Han

Abstract: Sparse recursive least squares (RLS) algorithms designed by introducing a sparse penalty (regularization) into the standard RLS cost function, have been proposed in the literature. Compared with the standard RLS, the sparse RLS achieves faster convergence and better performance under sparse systems. Even though, it includes in the updating equation an additional sparse term, which not only incurs extra complexity but also prevents the use of existing fast implementations such as the stable fast transversal filter (SFFT) algorithm. In this paper, we aim to reduce the complexity of the sparse RLS for promoting its practicability. To achieve the goal, the sparse updating term is analyzed and then approximated. With an approximated sparse updating term, the fast implementation is enabled for the sparse RLS, achieving complexity reduction. To demonstrate the feasibility of the proposed scheme, the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -RLS (as a typical sparse RLS algorithm) coupled with an approximated sparse updating term is proposed, leading to the selective zero-attracting SFTF(SZA-SFTF) algorithm. The SZA-SFTF has a complexity of order O(11N), compared with O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) for the original l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -RLS. In term of performance, simulations of sparse system identification showed the SZA-SFTF considerably outperforms the standard SFTF and achieves close performance to the exact l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -RLS.