Kernel Methods

A B M Shawkat Ali

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Data Mining

¤ DM or KDD (Knowledge Discovery in Databases)

Extracting previously unknown, valid, and actionable information crucial decisions

¤ Approach ModelTrain Data

crucial decisions

Test Data

History of SVM• The original optimal hyperplane algorithm proposed by Vladimir

Vapnik in 1963 was a linear classifier.

• However, in 1992, Bernhard Boser, Isabelle Guyon and Vapnik suggested a way to create non-linear classifiers by applying the kernel trick (originally proposed by Aizerman et al.) to maximum-margin hyperplanes. The resulting algorithm is formally similar, except that every dot product is replaced by a non-linear kernel function. This allows the algorithm to fit the maximum-margin hyperplane in a transformed feature space. The transformation may be non-linear and the transformed space high dimensional; thus though the classifier is a hyperplane in the high-dimensional feature space, it may be non-linear in the original input space.

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Property of the SVM

¤ Relatively new approach¤ Lot of interest recently: Many successes, e.g., text classification

¤ Important concepts: Transformation into high dimensional space Finding a "maximal margin" separation Structural risk minimization rather than Empirical risk minimization

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Support Vector Machine (SVM)

¤ Classification Grouping of similar data.

¤ Regression Prediction by historical knowledge.

¤ Novelty Detection To detect abnormal instances from a

dataset.¤ Clustering, Feature Selection

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SVM Block Diagram

Training Data Domain

Non linear Mapping by Kernel

To Choose Optimal Hyperplane

Linear Feature Space of SVM

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SVM Block Diagram

Constructed Model through Feature

knowledge

Class I

Class II

Test Data Domain

Kernel Mapping

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SVM Formulation

Siby ii ,1)( Xw

Si

iii y Xw

)(sign byySi

iii

XX

w

1 )(sign by Xw

ww

:min

1)( by ii Xw

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SVM Formulation

Siby ii ,1)( Xw

Si

iii y Xw

)(sign byySi

iii

XX

)(sign by Xw

iii

bbyC )(1:min

2

21

,Xww

w

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SVM Formulation

x X

)(

)(

)(

xX

xX

ii

),( K ),()()( xxxx ii K

)),((sign bKyySi

iii

xx

))()((sign byySi

iii

xx

)()( xxXX ii

Mercer’s Condition

)(sign byySi

iii

XX

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Types of Kernels

Common kernels for SVM¤ Linear ¤ Polynomial¤ Radial Basis Function

New kernels (not used in SVM)¤ Laplace ¤ Multiquadratic

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SVM kernel

dii kK )(),( xxxx

)exp(),( 22

).(2).().(

xxxxxxxx iii kkk

iK

Polynomial

Gaussian (Radial Basis Function)

Linear

xxxx iiK ),(

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Laplace kernel

Introduced by Pavel Paclik et. al. in Pattern Recognition letters 21 (2000)

Laplace Kernel based on Laplace Probability Density

N

i

D

j c

ijj

c h

xx

hNxf

1 1

exp2

11)|(ˆ

Smoothing Parameter (Sp)

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Linear Kernel

The reality of data separation

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RBF kernel

XOR solved by SVM

Input data x

Output class y

(-1,-1) -1

(-1,+1) +1

(+1,-1) +1

(+1,+1) -1

Table 5.3. Boolean XOR Problem

2)1(),( TjijiK xxxx

1 1

1 1 1 1 1 1.

1 1 1 1 1 1

1 1

Ti j

x x

•First, we transform the dataset by polynomial kernel as:

Here,

9111

1911

1191

1119

),( jiK xx

4 4 41

21 1 1i j

o y y Ki i i j i ji i j

x x

Therefore the kernel matrix is:

We can write the maximization term following SVM implementation given in Figure 5.20 as:

1 2 2 (9 2 2 2 91 2 3 4 1 1 2 1 3 1 4 222 22 2 9 2 9 )2 3 2 4 3 3 4 4

04321

4

1

i

iiy

1 2 3 40 , 0 , 0 , 0 subject to:

,

19

19

19

19

4321

4321

4321

4321

By solving these above equations we can write the solution to this optimisation problem as:

8

14321

.

Therefore, the decision function in the inner product representation is:

4

1

2

1

1125.0,ˆi

ii

n

iiii yKyf xxxxx

)()(

)(2)(2)())((2)(1

1)(2)(

)1),((),(

i

22112

2222112

11

22112

2211

2

jT

jijijijijiji

jijijiji

jiji

xxxxxxxxxxxx

xxxxxxxx

K

xx

xxxx

The 2nd degree polynomial kernel function:

Now we can write the 2nd degree polynomial transformation function as:

Tiiiiiii xxxxxx ]2,2,,2,,1[)( 21

2221

21 x

4

1

)(

i

iiio y x

)]()()()([8

14321 xxxx

0

0

0

2/1

0

0

2

2

1

2

1

1

2

2

1

2

1

1

2

2

1

2

1

1

2

2

1

2

1

1

8

1

0

2

2

2

1

0,0,0,2

1,0,0

2

1

22

21

21

x

x

x

xx

x

=

Therefore the optimal hyperplane function for this XOR problem is:

21)(ˆ xxf x

Conclusions

• Research Issues

– How to select a kernel automatically– How to select optimal parameter values for kernel