TY  - INPR
A1  - Blanchard, Gilles
A1  - Krämer, Nicole
T1  - Convergence rates of kernel conjugate gradient for random design regression
N2  - We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true
regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 8 
KW  - nonparametric regression
KW  - reproducing kernel Hilbert space
KW  - conjugate gradient
KW  - partial least squares
KW  - minimax convergence rates
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94195
SN  - 2193-6943
VL  - 5
IS  - 8
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Blanchard, Gilles
A1  - Mücke, Nicole
T1  - Optimal rates for regularization of statistical inverse learning problems
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 5 
KW  - statistical inverse problem
KW  - minimax rate
KW  - kernel method
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89782
SN  - 2193-6943
VL  - 5
IS  - 5
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  -