On the existence of global saturation for spectral regularization methods with optimal qualification

TL;DR

This paper investigates the conditions under which spectral regularization methods with optimal qualification exhibit global saturation, providing characterizations and examples to deepen understanding of their limitations.

Contribution

It establishes sufficient conditions for the existence of saturation in spectral regularization methods with optimal qualification and characterizes the saturation function and set.

Findings

Saturation exists under specific conditions on the regularization family.

Characterizations of saturation function and set are provided.

Examples illustrate the theoretical results.

Abstract

A family of real functions {g_\alpha} defining a spectral regularization method with optimal qualification is considered. Sufficient condition on the family and on the optimal qualification guaranteeing the existence of saturation are established. Appropriate characterizations of both the saturation function and the saturation set are found and some examples are provided.