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Instance Optimal Decoding and the Restricted Isometry Property

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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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Dates and versions

hal-01718411 , version 1 (27-02-2018)
hal-01718411 , version 2 (01-03-2018)

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Nicolas Keriven, Rémi Gribonval. Instance Optimal Decoding and the Restricted Isometry Property. 8th International Conference on New Computational Methods for Inverse Problems (NCMIP), May 2018, Cachan, France. pp.012002, ⟨10.1088/1742-6596/1131/1/012002⟩. ⟨hal-01718411v2⟩
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