Non-linear interpolation learning for example-based inverse problem regularization

Abstract

A large number of signal recovery problems are not well-posed-if not ill-posed-that require extra regularization to be tackled. In this context, the ability to inject physical knowledge is of utmost importance to design effective regularization schemes. However, most physically relevant models are generally nonlinear: signals generally lie on an unknown low-dimensional manifolds structure, which needs to be learnt. This is however quite challenging when available training samples are scarce. To that end, we investigate a novel approach that builds upon learning a non-linear interpolatory scheme from examples. We show how the proposed approach resonates with transportbased methods, but with a learnt metric. This eventually allows to build efficient non-linear regularizations for linear inverse problems. Extensive numerical experiments have been carried out to evaluate the performances of the proposed approach. We further illustrate its application to a blind regression problem in the field of γ-ray spectroscopy