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NeuroCOLT
Technical Report NC-TR-98-027
Generalization
Bounds and Learning Rates for Regularized Principal Manifolds
Alex J. Smola
GMD
Robert C. Williamson
ANU
Bernhard Schoelkopf
GMD
Received:
21-SEP-98
Abstract
We derive uniform convergence bounds and learning
rates for regularized principal manifolds. This builds on previous
work of Kegl et al., however we are able to obtain stronger bounds
taking advantage of the decomposition of the principal manifold in
terms of kernel functions. In particular, we are able to give bounds
on the covering numbers which are independent of the number of basis
functions (line elements) used. Finally we are able to obtain a nearly
optimal learning rate of order $O(m^{-1/2+/alpha})$ for certain types
of regularization operators, where m is the sample size and ff an
arbitrary positive constant. A companion paper [NC-TR-98-028]
describes the basic algorithm, details of the implementation and experimental
results.
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