Instance Optimal Decoding and the Restricted Isometry Property

Abstract : 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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https://hal.inria.fr/hal-01718411
Contributor : Nicolas Keriven <>
Submitted on : Tuesday, February 27, 2018 - 3:00:50 PM
Last modification on : Friday, September 13, 2019 - 9:50:02 AM
Long-term archiving on: Monday, May 28, 2018 - 10:30:59 AM

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  • HAL Id : hal-01718411, version 1
  • ARXIV : 1802.09905

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Nicolas Keriven, Rémi Gribonval. Instance Optimal Decoding and the Restricted Isometry Property. 2018. ⟨hal-01718411v1⟩

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