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

We study minimax estimation rates for a class of statistical inverse problems with non-linear operators acting pointwise. A general approach is applied to several cases motivated by concrete statistical applications. We derive new matching upper and lower bounds (up to logarithmic factors) on the risk for a pointwise function-dependent loss that captures spatial heterogeneity of the target function. The upper bound is obtained via easy to implement plug-in estimators based on hard-wavelet thresholding that are shown to be fully adaptive, both spatially and over H\"older smoothness classes. We consider a modified notion of H\"older smoothness scales that are natural in this setting and derive new estimates for the size of the wavelet coefficients of the non-linearly transformed function.