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Support

Vector

Method

for

Function

Approximation,

Regression

Estimation,

and

Signal

Processing·

Vladimir

Vapnik

AT&T

Research

101 Crawfords Corner

Holmdel, N J 07733

vlad@research.att.com

Steven

Golowich

Bell Laboratories

700

Mountain

Ave.

Murray

Hill, NJ 07974

golowich@bell-Iabs.com

Abstract

Alex

Smola·

GMD

First

Rudower Shausee 5

12489 Berlin

asm@big.att.com

The

Support

Vector (SV)

method

was recently proposed for es-

timating

regressions, constructing multidimensional splines,

and

solving linear

operator

equations [Vapnik, 1995]. In

this

presenta-

tion

report

results

applying

the

method

these problems.

Introduction

The

Support

Vector

method

is a universal

tool

for solving multidimensional function

estimation

problems. Initially it was designed

solve

pattern

recognition problems,

where in order

find a decision rule with good generalization ability one selects

some (small) subset

the

training

data

, called

the

Support

Vectors (SVs).

Optimal

separation

the

SV s is equivalent

optimal

separation

the

entire

data.

This

led

a new

method

representing decision functions where

the

decision

functions are a linear expansion on a basis whose elements are nonlinear functions

parameterized by

the

SVs (we need one SV for each element

the

basis).

This

type

function representation is especially useful for high dimensional

input

space:

the

number

free

parameters

in this representation is

equal

the

number

SVs

but

does

not

depend

the

dimensionality

the

space.

Later

the

method

was extended

real-valued functions.

This

allows us

expand

high-dimensional functions using a

small

basis constructed from SVs.

This

·smola@prosun.first.gmd.de

282

Vapnik,

Golowich

and

A. Smola

novel

type

function representation opens new opportunities for solving various

problems

function

approximation

and

estimation.

this

paper

demonstrate

that

using

the

SV technique one can solve problems

that

in classical techniques would require

estimating

a large number

free

param-

eters.

particular

construct one

and

two dimensional splines with

arbitrary

number

grid points. Using linear splines

approximate

non-linear functions.

We show

that

by reducing requirements on

the

accuracy

approximation, one de-

creases

the

number

SVs which leads

data

compression. We also show

that

the

SV technique

a useful tool for regression

estimation.

Lastly

demonstrate

that

using

the

SV function representation for solving inverse ill-posed problems provides

an additional

opportunity

for regularization.

method

for

estimation

real

functions

Let x E R

and

Y E

Rl.

Consider

the

following set

real functions: a vector x

mapped

into some a priori chosen Hilbert space, where

define functions

that

are

linear in

their

parameters

Y =

I(x,w)

= L

Wi<Pi(X),

W =

(WI,

...

,WN,

... ) E n

(1)

i=1

[Vapnik, 1995]

the

following

method

for

estimating

functions in

the

set (1) based

training

data

(Xl, Yd,

. ,

(Xl,

Yl)

was suggested: find

the

function

that

minimizes

the

following functional:

R(w) = £ L

IYi

- I(Xi,

w)lt:

+ I(w, w),

(2)

i=1

where

{

I(x,

w)1

£,

I(x,

w)lt:

I(x,

w)l-

£ otherwise,

(3)

(w,

the

inner

product

two vectors,

and

I is some constant.

was shown

that

the

function minimizing this functional

has

a form:

I(x,

a, a*) =

L(a;

ai)((xi),

(x))

+ b

(4)

;=1

where

ai,

2::

0 with

aiai

= 0

and

((Xi),

(x»

the inner

product

two

elements

Hilbert space.

To find

the

coefficients

and

ai one

has

solve

the

following quadratic optimiza-

tion problem: maximize

the

functional

ill

W(a*, a) =

-£

L(a;

+ai)+

Ly(a;

-ai)-~

L (a;

-ai)(aj

-aj

)((Xi),

(Xj)),

i=1

;=1

i,j=1

subject

constraints

L(ai-ai)=O,

O~ai,a;~C,

i=l,

...

,f.

i=1

(5)

(6)

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