The Bouligand Influence Function and its Application for Support Vector Machines

Arnout Van Messem

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
Title of host publication	15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium)
Publication status	Published - 18 Oct 2007

Publication series

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

Keywords

SVM
robustness
Bouligand

Cite this

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title = "The Bouligand Influence Function and its Application for Support Vector Machines",

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.",

keywords = "SVM, robustness, Bouligand",

author = "{Van Messem}, Arnout",

year = "2007",

month = oct,

day = "18",

language = "English",

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

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

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The Bouligand Influence Function and its Application for Support Vector Machines. / Van Messem, Arnout.
15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium). 2007. (15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium)).

Research output: Chapter in Book/Report/Conference proceeding › Meeting abstract (Book)

TY - CHAP

T1 - The Bouligand Influence Function and its Application for Support Vector Machines

AU - Van Messem, Arnout

PY - 2007/10/18

Y1 - 2007/10/18

N2 - 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.

AB - 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.

KW - SVM

KW - robustness

KW - Bouligand

M3 - Meeting abstract (Book)

T3 - 15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium)

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

ER -

The Bouligand Influence Function and its Application for Support Vector Machines

Abstract

Publication series

Keywords

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