NAMColloquium Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem |  |
Vortragende Person:Prof. Dr. Nicole Mücke, Technische Universität Braunschweig
Veranstaltungsort:Institut für Numerische und Angewandte Mathematik, Lotzestraße 16-18MN 55Gras Geo Map
Veranstalter:Institut für Numerische und Angewandte Mathematik
Beschreibung:
We consider the problem of learning a linear operator A 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 an inverse problem for A with the undesirable feature that its forward operator is
generally non-compact (even if A is assumed to be compact or of p-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated 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 proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.
generally non-compact (even if A is assumed to be compact or of p-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated 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 proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.
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Veranstaltungsart:Kolloquium
Veranstaltungssprache:Englisch
Kategorie:Forschung
Name der einladenden Person:Prof. Dr. Thorsten Hohage
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Direkter Link zur Veranstaltung:https://events.goettingen-campus.de/event?eventId=10216818
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