Minimax rates for statistical inverse problems under general source conditions

TL;DR

This paper establishes the minimax reconstruction rates for linear ill-posed problems in Hilbert spaces under general source conditions, extending classical results to broader smoothness classes.

Contribution

It generalizes the minimax rate analysis to source sets beyond ellipsoids, providing explicit risk formulas for truncated series estimators in ill-posed inverse problems.

Findings

Minimax rates are characterized for general source conditions.

Explicit risk formulas for truncated series estimators are derived.

Several examples illustrate the theoretical results.

Abstract

We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R.~C. Liu, and B. MacGibbon, {\it Minimax risk over hyperrectangles, and implications}, Ann.~ Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.