Smoothed residual stopping for statistical inverse problems via truncated SVD estimation

TL;DR

This paper investigates an early stopping rule based on smoothed residuals for truncated SVD estimation in inverse problems, providing bounds for adaptivity and demonstrating the method's effectiveness through simulations.

Contribution

It introduces a smoothed residual stopping rule that enables adaptivity in truncated SVD estimation for inverse problems, with theoretical risk bounds and practical validation.

Findings

Moderate smoothing allows adaptation over various signal smoothness classes.

Oversmoothing of residuals leads to suboptimal convergence rates.

Monte-Carlo simulations confirm theoretical results.

Abstract

This work examines under what circumstances adaptivity for truncated SVD estimation can be achieved by an early stopping rule based on the smoothed residuals $ \| ( A A^{\top} )^{\alpha / 2} ( Y - A \hat{\mu}^{( m )}) \|^{2} $. Lower and upper bounds for the risk are derived, which show that moderate smoothing of the residuals can be used to adapt over classes of signals with varying smoothness, while oversmoothing yields suboptimal convergence rates. The theoretical results are illustrated by Monte-Carlo simulations.