An Online Parallel Algorithm for Recursive Estimation of Sparse Signals

In this paper, we consider a recursive estimation problem for linear regression where the signal to be estimated admits a sparse representation and measurement samples are only sequentially available. We propose a convergent parallel estimation scheme that consists of solving a sequence of ℓ
₁
-regularized least-square problems approximately. The proposed scheme is novel in three aspects: 1) all elements of the unknown vector variable are updated in parallel at each time instant, and the convergence speed is much faster than state-of-the-art schemes which update the elements sequentially; 2) both the update direction and stepsize of each element have simple closed-form expressions, so the algorithm is suitable for online (real-time) implementation; and 3) the stepsize is designed to accelerate the convergence but it does not suffer from the common intricacy of parameter tuning. Both centralized and distributed implementation schemes are discussed. The attractive features of the proposed algorithm are also illustrated numerically.