Instance Optimal Decoding and the Restricted Isometry Property
This work addresses theoretical guarantees for robust decoding in inverse problems, which is incremental as it builds on compressive sensing concepts to handle non-linear cases.
The paper tackles the problem of information preservation in ill-posed, non-linear inverse problems by establishing necessary and sufficient conditions for an instance optimal decoder that is robust to noise and modeling error, using a non-linear Lower Restricted Isometry Property (LRIP) and extending results to random measurement operators and neural net invertibility.
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.