Sparse system identification by exponentially weighted adaptive parallel projection and generalized soft-thresholding

In this paper, we propose a novel online scheme named the adaptive proximal forward-backward splitting method to suppress the sum of'smooth' and'nonsmooth' convex functions, both of which are assumed time-varying. We derive a powerful algorithm for sparse system identification by defining each function as a certain average squared distance ('smooth') and a weighted ℓ1-norm ('nonsmooth'). The smooth term brings an exponentially weighted average of the metric projections of the current estimate onto linear varieties, of which the number grows as new measurements arrive. The presented recursive formula realizes an efficient computation of the average (the exponentially weighted adaptive parallel projection), which contributes the fast and stable convergence. The nonsmooth term, on the other hand, brings the weighted soft-thresholding, contributing the enhancement of the filter sparsity. The weights are adaptively controlled according to the filter coefficients so that the softthresholding gives significant impacts solely to inactive coefficients (coefficients close to zero). The numerical example demonstrates the efficacy of the proposed algorithm.