Regularization independent of the noise level: an analysis of quasi-optimality

TL;DR

This paper analyzes the quasi-optimality criterion for inverse problems, showing it can be rate-optimal on average without knowing the noise level, and demonstrates its effectiveness with a finance calibration example.

Contribution

It provides an average case analysis proving quasi-optimality can be noise-level independent and rate-optimal on average for spectral cut-off estimators.

Findings

Quasi-optimality is rate-optimal on average.

The method performs well in a finance calibration problem.

It works without explicit noise level knowledge.

Abstract

The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators and we prove that the quasi-optimality criterion determines estimators which are rate-optimal {\em on average}. Its practical performance is illustrated with a calibration problem from mathematical finance.