8978
2016
2016
eng
36
5
5
preprint
Universitätsverlag Potsdam
Potsdam
1
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Optimal rates for regularization of statistical inverse learning problems
We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.
urn:nbn:de:kobv:517-opus4-89782
2193-6943 (online)
Keine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Gilles Blanchard
Nicole Mücke
Preprints des Instituts für Mathematik der Universität Potsdam
5 (2016) 5
eng
uncontrolled
statistical inverse problem
eng
uncontrolled
minimax rate
eng
uncontrolled
kernel method
Mathematik
Minimax procedures
Estimation
Asymptotic properties
Inverse problems
open_access
Institut für Mathematik
Universitätsverlag Potsdam
2016
Universität Potsdam
Universitätsverlag Potsdam
https://publishup.uni-potsdam.de/files/8978/premath05.pdf