Adaptive Diffusion Pairwise Fused Lasso LMS Algorithm Over Networks

The topic of identification for sparse vector in a distributed way has triggered great interest in the area of adaptive filtering. Grouping components in the sparse vector has been validated to be an efficient way for enhancing identification performance for sparse parameter. The technique of pairwise fused lasso, which can promote similarity between each possible pair of nonnegligible components in the sparse vector, does not require that the nonnegligible components have to be distributed in one or multiple clusters. In other words, the nonnegligible components may be randomly scattered in the unknown sparse vector. In this article, based on the technique of pairwise fused lasso, we propose the novel pairwise fused lasso diffusion least mean-square (PFL-DLMS) algorithm, to identify sparse vector. The objective function we construct consists of three terms, i.e., the mean-square error (MSE) term, the regularizing term promoting the sparsity of all components, and the regularizing term promoting the sparsity of difference between each pair of components in the unknown sparse vector. After investigating mean stability condition of mean-square behavior in theoretical analysis, we propose the strategy of variable regularizing coefficients to overcome the difficulty that the optimal regularizing coefficients are usually unknown. Finally, numerical experiments are conducted to verify the effectiveness of the PFL-DLMS algorithm in identifying and tracking sparse parameter vector.

I. Introduction

In many practical systems, the sparse impulse responses are often encountered, i.e., the signal vectors representing such responses consist of only a few nonnegligible components, while the other components are zero. Typical examples include spectrum sensing in cognitive radio [1], imaging denoising [2], and comparative genomic hybridization [3]. However, conventional adaptive algorithms for system identification, which include the least mean-square (LMS) algorithm [4] and the recursive least square (RLS) algorithm [5], do not consider the a priori information of sparsity in the cost function. Exploiting the a priori information about the sparsity of these vectors properly when designing adaptive algorithms has proven to lead to significant improvement for identifying sparse impulsive response [6], [7], [9]. In these works, the - or -norm of parameter estimate is incorporated into the mean-square error (MSE) objective function. More recently, inspired by superiority of distributed estimation over noncooperated estimation [10], [13], the study on identifying sparse vector has been generalized to the context of distributed adaptive estimation. Typical examples include the sparse incremental LMS algorithm [8], the sparse diffusion LMS algorithm [21], [22], and the proportionate diffusion LMS (PDLMS) algorithm [23]. Compared to the incremental-type [14], [15] distributed estimation, the diffusion-type [16]–[18] distributed estimation is more frequently used due to its power learning ability and strong robustness. Therefore, we focus on the diffusion-type distributed estimation for sparse vector in this article.