A Probabilistic Oracle Inequality and Quantification of Uncertainty of a modified Discrepancy Principle for Statistical Inverse Problems

TL;DR

This paper develops a probabilistic framework for a modified discrepancy principle in statistical inverse problems, providing oracle inequalities and uncertainty quantification, and compares it with existing methods both theoretically and numerically.

Contribution

It introduces a new adaptive spectral cut-off estimator with probabilistic oracle inequalities and uncertainty quantification, enhancing the classical discrepancy principle for inverse problems.

Findings

The new method achieves competitive or improved accuracy compared to existing methods.

Probabilistic oracle inequalities are established for the proposed estimator.

Numerical experiments demonstrate the effectiveness of the method.

Abstract

In this note we consider spectral cut-off estimators to solve a statistical linear inverse problem under arbitrary white noise. The truncation level is determined with a recently introduced adaptive method based on the classical discrepancy principle. We provide probabilistic oracle inequalities together with quantification of uncertainty for general linear problems. Moreover, we compare the new method to existing ones, namely early stopping sequential discrepancy principle and the balancing principle, both theoretically and numerically.