Nonlinear compressed sensing based on composite mappings and its pointwise linearization

TL;DR

This paper introduces a novel nonlinear compressed sensing framework based on composite mappings and a pointwise linearization technique, enabling exact sparse signal recovery despite nonlinear measurement distortions.

Contribution

It proposes a new perspective of nonlinear CS via composite mappings and develops a pointwise linearization method to facilitate recovery using linear CS algorithms.

Findings

Derived conditions for exact sparse recovery with nonlinear composite mappings

Introduced a pointwise linearization method for nonlinear measurement models

Ensured exact recovery of sparse signals even when the nonlinear mapping is not injective

Abstract

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 Phi, i.e., Phi(x) = F(Ax) and Phi(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 Phi(x). Besides, we also provide a special pointwise linearization method, which can turn the nonlinear composite mapping Phi, 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 Phi 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.