Compressed Sensing with Nonlinear Observations

Compressed sensing is a recently developed signal acquisition technique. In contrast to traditional sampling methods, significantly fewer samples are required whenever the signals admit a sparse representation. Crucially, sampling methods can be constructed that allow the reconstruction of sparse signals from a small number of measurements using efficient algorithms. We have recently generalised these ideas in two important ways. We have developed methods and theoretical results that allow much more general constraints to be imposed on the signal and we have also extended the approach to more general Hilbert spaces. In this paper we introduce a further generalisation to compressed sensing and allow for non-linear sampling methods. This is achieved by using a recently introduced generalisation of the Restricted Isometry Property (or the bi-Lipschitz condition) traditionally imposed on the compressed sensing system. We show that, if this more general condition holds for the nonlinear sampling system, then we can reconstruct signals from non-linear compressive measurements.