Low complexity projection-based adaptive algorithm for sparse system identification and signal reconstruction

The present paper introduces a low complexity online convex analytic tool for time-varying sparse system identification and signal reconstruction tasks. The available information enters the design in two ways; (i) the sequentially arriving training data generate a sequence of simple closed convex sets, namely hyperslabs, and (ii) the information regarding the cardinality of the support of the unknown system/signal is used to create another sequence of closed convex sets, namely weighted ℓ₁-balls. In such a way, searching for the unknown system/signal becomes the task of solving a convex feasibility problem with an infinite number of constraints. The basic tool to solve such a problem, with computational load that scales linearly to the number of unknowns, is the projection onto a closed convex set, and more importantly the subgradient projection mapping associated to a convex function. A convergence analysis of the proposed algorithm is given based on very recent advances of projection-based adaptive algorithms, and numerical results are presented to support the introduced theory.