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.
Highlights
Inverse problems are ubiquitous in all areas of data science
Our main results show that the existence of a decoder that is robust to noise and modelling error is equivalent to a modified Restricted Isometry Property (RIP), which is a classical property in compressive sensing [9]
In this paper we generalized a classical property, the equivalence between the existence of an instance-optimal decoder and the Lower -RIP (LRIP), to non-linear inverse problems with possible quantization error or limited algorithmic precision, and data that live in any pseudometric set
Summary
Inverse problems are ubiquitous in all areas of data science. While linear inverse problems have been arguably far more studied in the literature, some frameworks are intrinsically non-linear [14]. Our main results show that the existence of a decoder that is robust to noise and modelling error is equivalent to a modified Restricted Isometry Property (RIP), which is a classical property in compressive sensing [9]. Our goal is to characterize necessary and sufficient conditions for the existence of an instance optimal decoder for the problem (1). The interplay between the two notions was in particular studied in [12] in the finitedimensional case These results were later extended to more general models in [21], and to any linear measurement operators in [8], which is the main inspiration behind the present work. We distinguish the case where the operator Ψ is deterministic, or, equivalently, when it is random but one seeks so-called uniform recovery guarantees, and the case of non-uniform recovery
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