# The Bouligand Influence Function and its Application for Support Vector Machines

Arnout Van Messem

Research output: Chapter in Book/Report/Conference proceedingMeeting abstract (Book)

## Abstract

We propose the Bouligand influence function (BIF) as a
new concept for robust statistics. The BIF is a modification of F.R.
Hampel's influence function (IF) and is based on a special cone
derivative instead of the usual G{\^a}teaux-derivative. If the BIF
does exist, then the IF does also exist and both are equal. The
usefulness of Bouligand-derivatives to robust statistics is
explained.

In the second part of the talk we apply the BIF to support vector
machines based on a non-smooth loss function for which the influence
was unknown. We show for the regression case that many support
vector machines based on a Lipschitz continuous loss function and a
bounded kernel have a bounded BIF and hence also have a bounded IF.
In this respect such SVMs are therefore robust. Special cases are
SVMs based on the $\epsilon$-insensitive loss, Huber's loss, and
kernel based quantile regression based on the pinball loss.
Original language English 15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium) Published - 18 Oct 2007

### Publication series

Name 15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium)

• SVM
• robustness
• Bouligand

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