5,053 research outputs found
Qualitative Robustness of Support Vector Machines
Support vector machines have attracted much attention in theoretical and in
applied statistics. Main topics of recent interest are consistency, learning
rates and robustness. In this article, it is shown that support vector machines
are qualitatively robust. Since support vector machines can be represented by a
functional on the set of all probability measures, qualitative robustness is
proven by showing that this functional is continuous with respect to the
topology generated by weak convergence of probability measures. Combined with
the existence and uniqueness of support vector machines, our results show that
support vector machines are the solutions of a well-posed mathematical problem
in Hadamard's sense
Fast rates for support vector machines using Gaussian kernels
For binary classification we establish learning rates up to the order of
for support vector machines (SVMs) with hinge loss and Gaussian RBF
kernels. These rates are in terms of two assumptions on the considered
distributions: Tsybakov's noise assumption to establish a small estimation
error, and a new geometric noise condition which is used to bound the
approximation error. Unlike previously proposed concepts for bounding the
approximation error, the geometric noise assumption does not employ any
smoothness assumption.Comment: Published at http://dx.doi.org/10.1214/009053606000001226 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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