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.
Type de document :
Pré-publication, Document de travail
2018
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https://hal.inria.fr/hal-01718411
Contributeur : Nicolas Keriven <>
Soumis le : jeudi 1 mars 2018 - 11:30:17
Dernière modification le : mercredi 16 mai 2018 - 11:24:14
Document(s) archivé(s) le : mercredi 30 mai 2018 - 12:57:32

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  • HAL Id : hal-01718411, version 2
  • ARXIV : 1802.09905

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Nicolas Keriven, Rémi Gribonval. Instance Optimal Decoding and the Restricted Isometry Property. 2018. 〈hal-01718411v2〉

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