Convergence of generalized cross-validation with applications to ill-posed integral equations

TL;DR

This paper proves that generalized cross-validation reliably selects parameters for solving ill-posed inverse problems, achieving optimal error bounds without requiring the true solution to be self-similar, with demonstrated numerical convergence.

Contribution

It establishes the non-asymptotic optimality of GCV for polynomially ill-posed problems without self-similarity assumptions.

Findings

GCV achieves order-optimal error bounds with high probability.

Numerical experiments confirm convergence in image deblurring and CT reconstruction.

GCV is effective without the need for self-similarity conditions.

Abstract

In this article, we rigorously establish the consistency of generalized cross-validation as a parameter-choice rule for solving inverse problems. We prove that the index chosen by leave-one-out GCV achieves a non-asymptotic, order-optimal error bound with high probability for polynomially ill-posed compact operators. Hereby it is remarkable that the unknown true solution need not satisfy a self-similarity condition, which is generally needed for other heuristic parameter choice rules. We quantify the rate and demonstrate convergence numerically on integral equation test cases, including image deblurring and CT reconstruction.