Instance Optimal Decoding and the Restricted Isometry Property

Nicolas Keriven (1) , Rémi Gribonval (2)

1 ENS-PSL - École normale supérieure - Paris
2 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio

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.

Mots clés: Random features; Compressed Sensing; Non-linear inverse problem

Domaines: Théorie de l'information [cs.IT]; Apprentissage [cs.LG]

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