  In this paper, we address the question of information preservation in
ill-posed, non-linear inverse problems, assuming that the measured data is
close to a low-dimensional model set. We provide necessary and sufficient
conditions for the existence of a so-called instance optimal decoder, i.e.,
that is robust to noise and modelling error. Inspired by existing results in
compressive sensing, our analysis is based on a (Lower) Restricted Isometry
Property (LRIP), formulated in a non-linear fashion. We also provide sufficient
conditions for non-uniform recovery with random measurement operators, with a
new formulation of the LRIP. We finish by describing typical strategies to
prove the LRIP in both linear and non-linear cases, and illustrate our results
by studying the invertibility of a one-layer neural net with random weights.
