Smoothed residual stopping for statistical inverse problems via truncated SVD estimation

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 $\|(AA^{\top })^{\alpha /2}(Y-A\widehat{\mu }^{(m)})\|^{2}$. 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 range of smoothness classes for which adaptation is possible can be controlled via $\alpha $. The theoretical results are illustrated by Monte-Carlo simulations.