Online Sparse System Identification and Signal Reconstruction Using Projections Onto Weighted $\ell_{1}$ Balls

This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab is the geometric equivalent of a cost criterion, that quantifies “data mismatch.” Sparsity is imposed by the introduction of appropriately designed weighted ℓ
₁
balls and the related projection operator is also derived. The algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted ℓ
₁
balls. The resulting scheme exhibits linear dependence, with respect to the unknown system's order, on the number of multiplications/additions and an O(Llog
₂
L) dependence on sorting operations, where L is the length of the system/signal to be estimated. Numerical results are also given to validate the performance of the proposed method against the Least-Absolute Shrinkage and Selection Operator (LASSO) algorithm and two very recently developed adaptive sparse schemes that fuse arguments from the LMS/RLS adaptation mechanisms with those imposed by the lasso rational.