Instance Optimal Decoding and the Restricted Isometry Property

Abstract : 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.
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Submitted on : Thursday, March 1, 2018 - 11:30:17 AM
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Nicolas Keriven, Rémi Gribonval. Instance Optimal Decoding and the Restricted Isometry Property. International Conference on New Computational Methods for Inverse Problems (NCMIP), May 2018, Cachan, France. pp.1-12, ⟨10.1088/1742-6596/1131/1/012002⟩. ⟨hal-01718411v2⟩

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