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Radial-Basis FunctionBy Mrs. Pritam S. Salankar 

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Introduction

What is RBF networks? A kind of superised neural networks

!esi"n of #eural #etworks as cure fittin" $appro%imation& pro'lem

(earnin"

Find a surface that 'est fits to "ien trainin" data

)enerali*ation

use of this multidimensional surface to interpolate test data

X

X

XXXX OX

TrainingData

Generalization

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Structure of RBF Networks

+nput layer 

,idden layer,idden units proide a set of 'asis function

,i"h dimension more linearly separa'le /oer0s 1heorem2

3utput layer (inear com'ination of hidden functions

...

x1x2x3

X

...

Increase

ddemesion:N->m1  Output!oreli"el to#elinearlseperated

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Cover’s Theorem on Separability!"

 A pattern classification cast in hi"h dimensional spacenonlinearly is more likely to 'e linearly separa'le than in alow dimension space

X

O

O

O

X

X

X

X

XO

X O

O

OX 

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Cover’s Theorem on Separability#"

/oer0s theorem in case of polynomial functions(et

 

Polynomial hidden function

r-th order rational arieties

Pro'a'ility that particular dichotomy picked at random is

# 4 of data points

Separatin" surface de"ree of freedom +f is increasin"5 the pro'a'ility of separa'ility is increasin"

...

... 

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$%ample& '(R )roblem

63R data are not linearly separa'le

#onlinear transformation

(inearly separa'le in spaceInput patternX irst !i""enunction#econ"!i""enuncti

%% $1%'

& &

&

$1&

'$%&1'$1&%'Decision  (oundar

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Separability Capacity of Surface

Separatin" capacity of surface #7%pected ma%imum num'er of ectors linearly separa'le in a space

of dimensionality

Surface of hi"h dimension has hi"h separatin" capacity

/orollary to /oer0s theorem27#28 9

Median#289

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Interpolation )roblem!"

+nterpolation pro'lem"ien a set of # different points and a

correspondin" set of # real num'er 5 find afunction

that satisfies this condition

Radial-'asis functionFunction Form

w1−=== NN  N  N  N 321 ,, %:

%9

d

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Interpolation )roblem#"

Micchelli0s 1heorem +f 'e a set of distinct points5 then #-'y-# interpolation matri%

is nonsin"ular

1ypes of RBF functionMulti;uadrics

+nerse multi;uadrics

)aussian functions

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Interpolation )roblem*"

7%ample<$%5d&= 8< $-:5:&5$>59&5$:5:& =..............3

%-1 11

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Summary

RBF #etworks1hree layer +nput layer5 hidden layer5 output layer 

,idden units proide set of 'asis functions for input ectors

!imension of hidden layer is much lar"er than that of input layer 

3utput is o'tained from linear com'ination of hidden functions

/oer0s theorem A pattern classification cast in hi"h dimensional space nonlinearly is

more likely to 'e linearly separa'le than in a low dimension space

+nterpolation pro'lemFind a function which satisfies

se radial 'asis function 5

+f 'e a set of distinct points5 then solution e%istsMicchelli0s 1heorem2