Inverse Problems and Machine Learning

Date: October 6, 2021

Time: 12:40PM - 1:40PM

Location: Zoom

Speaker: Sergio Brenner Miguel, Universität Heidelberg

Title: Multiplicative Deconvolution in nonparametric estimations

Abstract: In this talk we are interested in the estimation of an unknown density f with support on the positive real line based on an i.i.d. sample with multiplicative measurement errors. Using the rich theory of Mellin transform we identify and study the underlying inverse problem. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the density f, a regularisation of the inverse of the Mellin transform by a spectral cut-off and a data-driven model selection in order to deal with the upcoming bias-variance trade-off. We introduce and discuss further the Mellin-Sobolev spaces which characterize the regularity of the unknown density f through the decay of its Mellin transform. Additionally we show minimax-optimality over Mellin-Sobolev spaces of the data-driven density estimator and hence its adaptivity.