A new QR decomposition-based RLS algorithm using the split Bregman method for L1-regularized problems

• A new adaptive split Bregman algorithm for L1-regularized problems.
• QR decomposition-based RLS algorithm for implementation.
• Regularization parameter selection formulae for sparsity control.
• Multivariate version for sparse principal component analysis.

The split Bregman (SB) method can solve a broad class of L1-regularized optimization problems and has been widely used for sparse signal processing in a variety of applications. To achieve lower complexity and to cope with time-varying environments, we develop a new adaptive version of the SB method for finding online sparse solutions. This algorithm is derived from the recursive least squares (RLS) optimization problem, where the SB method is used to separate the regularization term from the constrained optimization. This algorithm is numerically more stable and easily amenable to multivariate implementation due to the use of a QR decomposition (QRD) structure. An efficient method is further developed for selecting the thresholding rule, which controls the sparsity level of the estimator. Moreover, the SB-QRRLS algorithm is extended to a multivariate version to solve the sparse principal component analysis (SPCA) problem. Simulation results are presented to illustrate the effectiveness of the proposed algorithms in sparse system estimation and SPCA. We show that the convergence and tracking performance of the proposed algorithms compares favorably with conventional algorithms.