Nonlinear compressed sensing with application to phase retrieval

We extend the ideas of compressed sensing to nonlinear measurement systems. In particular, we treat the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We derive several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. We then specialize our algorithms to the problem of phase retrieval and develop an efficient method for retrieving a signal from its magnitude only measurements.