Optimal Rates of Convergence for Nonparametric Statistical Inverse Problems

Nonparametric statistical inverse problems

We explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems. Simple examples are used to present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence, as well as more recent concepts such as adaptive estimation, oracle inequalities, modern model selection methods, Stein’s unb...

Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications

During the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators...

Convergence rates of spectral regularization methods for statistical inverse learning problems

Consider an inverse problem of the form g = Af , where A is a known operator between Hilbert function spaces, and assume that we observe g at some randomly drawn points X1, ..., Xn which are i.i.d. according to some distribution PX , and where additionally each observation is subject to a random independent noise. The goal is to recover the function g. Here it is assumed that for each point x t...

Optimal Rates of Convergence to Bayes Risk in Nonparametric Discrimination

Convergence rates for the Bayesian approach to linear inverse problems∗

Recently, the metrics of Ky Fan and Prokhorov were introduced as a tool for studying convergence in stochastic ill-posed problems. In this work, we show that the Bayesian approach to linear inverse problems can be examined in the new framework as well. We consider the finitedimensional case where the measurements are disturbed by an additive normal noise and the prior distribution is normal. Co...