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Consistency and Robustness Properties of Support Vector Machines for Heavy-Tailed Distributions
Van Messem, Arnout; Christmann, Andreas
2010
 

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Abstract :
[en] Recently results on consistency and robustness of support vector machines (SVMs) were derived for non-negative convex losses L of Nemitski type given some weak moment condition for the joint distribution P on X × Y (Christmann and Steinwart, 2007). However, this condition excludes heavy-tailed distributions such as the Cauchy distribution or several extreme value distributions. The condition on P can be weakened to only a condition on the marginal distribution PX by shifting the loss L downwards, a trick that will enlarge the applicability of SVMs to heavy-tailed conditional distributions. More precisely, we define the shifted loss L⋆(x,y,f(x)) := L(x,y,f(x))− L(x,y,0). Obviously, this new “loss” L⋆(x,y,f(x)) can be negative. We define the decision function of the shifted SVM as fL⋆,P,λ. We will give some properties of L⋆ and fL⋆,P,λ and we will state a representer theorem and results on both risk-consistency as well as consistency of the solution. Finally we will show that, given some conditions, fL⋆,P,λ is robust in the sense of both Hampel’s influence function as well as the Bouligand influence function introduced by Christmann and Van Messem (2008).
Disciplines :
Mathematics
Author, co-author :
Van Messem, Arnout  ;  Vrije Universiteit Brussel - VUB > Departement Wiskunde
Christmann, Andreas
Language :
English
Title :
Consistency and Robustness Properties of Support Vector Machines for Heavy-Tailed Distributions
Publication date :
28 May 2010
Event name :
VUB PhD Day
Event organizer :
Vrije Universiteit Brussel
Event place :
Brussel, Belgium
Event date :
28-05-2010
Available on ORBi :
since 13 October 2021

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