Minimax theory for a class of non-linear statistical inverse problems

TL;DR

This paper introduces a two-step method for solving non-linear statistical inverse problems, combining data smoothing and wavelet thresholding, achieving near-minimax optimality in a pointwise function-dependent framework.

Contribution

It proposes a novel two-step approach that reduces non-linear inverse problems to linear ones and demonstrates near-minimax optimality using a new smoothness scale.

Findings

Wavelet thresholding achieves near-minimax optimal noise reduction.

The method effectively linearizes non-linear inverse problems.

Analysis introduces a new notion of H"older smoothness scales.

Abstract

We study a class of statistical inverse problems with non-linear pointwise operators motivated by concrete statistical applications. A two-step procedure is proposed, where the first step smoothes the data and inverts the non-linearity. This reduces the initial non-linear problem to a linear inverse problem with deterministic noise, which is then solved in a second step. The noise reduction step is based on wavelet thresholding and is shown to be minimax optimal (up to logarithmic factors) in a pointwise function-dependent sense. Our analysis is based on a modified notion of H\"older smoothness scales that are natural in this setting.