Non-convexly constrained linear inverse problems

TL;DR

This paper investigates the stable inversion of ill-posed linear operators by enforcing solutions within non-convex sets, extending sparsity-based methods from finite dimensions to Hilbert spaces with theoretical guarantees and iterative algorithms.

Contribution

It introduces a framework for regularizing ill-posed inverse problems using non-convex constraints in Hilbert spaces, with theoretical analysis and an iterative algorithm.

Findings

Theoretical properties for stable inversion are established.

An iterative algorithm similar to the projected Landweber method is studied.

Extension of sparsity-based inversion techniques to non-convex constraints in Hilbert spaces.

Abstract

This paper considers the inversion of ill-posed linear operators. To regularise the problem the solution is enforced to lie in a non-convex subset. Theoretical properties for the stable inversion are derived and an iterative algorithm akin to the projected Landweber algorithm is studied. This work extends recent progress made on the efficient inversion of finite dimensional linear systems under a sparsity constraint to the Hilbert space setting and to more general non-convex constraints.