Nonlinear compressed sensing based on composite mappings and its pointwise linearization

Classical compressed sensing (CS) allows us to recover structured signals from far few linear measurements than traditionally prescribed, thereby efficiently decreasing sampling rates. However, if there exist nonlinearities in the measurements, is it still possible to recover sparse or structured signals from the nonlinear measurements? The research of nonlinear CS is devoted to answering this question. In this paper, unlike the existing research angles of nonlinear CS, we study it from the perspective of mapping decomposition, and propose a new concept, namely, nonlinear CS based on composite mappings. Through the analysis of two forms of a nonlinear composite mapping Φ, i.e., Φ(x) = F(Ax) and Φ(x) = AF(x), we give the requirements respectively for the sensing matrix A and the nonlinear mapping F when reconstructing all sparse signals exactly from the nonlinear measurements Φ(x). Besides, we also provide a special pointwise linearization method, which can turn the nonlinear composite mapping Φ, at each point in its domain, into an equivalent linear composite mapping. This linearization method can guarantee the exact recovery of all given sparse signals even if Φ is not an injection for all sparse signals. It may help us build an algorithm framework for the composite nonlinear CS in which we can take full advantage of the existing recovery algorithms belonging to linear CS.