  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.
