New operators for fixed-point theory: The sparsity-aware learning case

The present paper offers a link between fixed point theory and thresholding; one of the key enablers in sparsity-promoting algorithms, associated mostly with non-convex penalizing functions. A novel family of operators, the partially quasi-nonexpansive mappings, is introduced to provide the necessary theoretical foundations. Based on such fixed point theoretical ground, and motivated by hard thresholding, the generalized thresholding (GT) mapping is proposed that encompasses hard, soft, as well as recent advances of thresholding rules. GT is incorporated into an online/time-adaptive algorithm of linear complexity that demonstrates competitive performance with respect to computationally thirstier, state-of-the-art, RLS- and proportionate-type sparsity-aware methods.