  This paper studies a sparse signal recovery task in time-varying
(time-adaptive) environments. The contribution of the paper to sparsity-aware
online learning is threefold; first, a Generalized Thresholding (GT) operator,
which relates to both convex and non-convex penalty functions, is introduced.
This operator embodies, in a unified way, the majority of well-known
thresholding rules which promote sparsity. Second, a non-convexly constrained,
sparsity-promoting, online learning scheme, namely the Adaptive
Projection-based Generalized Thresholding (APGT), is developed that
incorporates the GT operator with a computational complexity that scales
linearly to the number of unknowns. Third, the novel family of partially
quasi-nonexpansive mappings is introduced as a functional analytic tool for
treating the GT operator. By building upon the rich fixed point theory, the
previous class of mappings helps us, also, to establish a link between the GT
operator and a union of linear subspaces; a non-convex object which lies at the
heart of any sparsity promoting technique, batch or online. Based on such a
functional analytic framework, a convergence analysis of the APGT is provided.
Furthermore, extensive experiments suggest that the APGT exhibits competitive
performance when compared to computationally more demanding alternatives, such
as the sparsity-promoting Affine Projection Algorithm (APA)- and Recursive
Least Squares (RLS)-based techniques.
