Session I.6 - Mathematical Foundations of Data Assimilation and Inverse Problems
Tuesday, June 13, 17:00 ~ 17:30
Learning linear operators: infinite-dimensional regression as an inverse problem
Mattes Mollenhauer
Freie Universität Berlin, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the problem of learning a linear operator between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as a statistical inverse problem with unknown noncompact forward operator. However, we prove that, in terms of spectral properties, this problem is equivalent to the well-known compact inverse problem with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under Hölder-type source conditions. The rates holds for a variety of relevant scenarios in functional regression and nonparametric regression with operator-valued kernels and match those of classical kernel regression with scalar response.
The preprint is available under https://arxiv.org/abs/2211.08875
Joint work with Nicole Mücke (TU Braunschweig, Germany) and Tim Sullivan (U Warwick, UK).