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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

Joint work with Nicole Mücke (TU Braunschweig, Germany) and Tim Sullivan (U Warwick, UK).

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