Zero-Attracting Recursive Least Squares Algorithms

Xia Hong, Junbin Gao, Sheng Chen

Abstract: The l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm sparsity constraint is a widely used technique for constructing sparse models. In this paper, two zeroattracting recursive least squares algorithms, which are referred to as ZA-RLS-I and ZA-RLS-II, are derived by employing the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm of the parameter vector constraint to facilitate model sparsity. To achieve a closed-form solution, the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm of the parameter vector is approximated by an adaptively weighted l2-norm in which the weighting factors are set as the inversion of the associated l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm of parameter estimates that are readily available in the adaptive learning environment. ZA-RLS-II is computationally more efficient than ZA-RLS-I by exploiting the known results from linear algebra and the sparsity of the system. The proposed algorithms are proven to converge, and adaptive sparse channel estimation is used to demonstrate the effectiveness of the proposed approach.