Instance Optimal Decoding and the Restricted Isometry Property
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In this paper, we study the preservation of information in ill-posed non-linear inverse problems, where the measured data is assumed to live near 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 a characterization for non-uniform recovery when the encoding process is randomly drawn, 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 network with random weights.
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