Regularization independent of the noise level: an analysis of quasi-optimality
Provides theoretical justification for a practical heuristic used in inverse problems, addressing a known gap between theory and practice.
The paper analyzes the quasi-optimality criterion for choosing regularization parameters in inverse problems without noise level, proving it is rate-optimal on average for spectral cut-off estimators, and demonstrates its effectiveness on a calibration problem from mathematical finance.
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.