Margin - University of Nottingham

Report
An Introduction to Support
Vector Machines
Presenter: Celina Xia
University of Nottingham
G54DMT - Data Mining Techniques and Applications
Outline
Maximizing the Margin
Linear SVM and Linear Separable
Case
Primal Optimization Problem
Dual Optimization Problem
Non-Separable Case
Non-Linear Case
Kernel Functions
Applications
G54DMT - Data Mining Techniques and Applications
Margin
Any of these
separating lines
would be fine..
..but which is
best?
G54DMT - Data Mining Techniques and Applications
Margin
Margin: the width that the boundary could
be increased by before hitting a datapoint.
margin
margin
Wide margin
Decision boundary
Narrow margin
G54DMT - Data Mining Techniques and Applications
SVMs reckon…
Decision boundary
The decision boundary
with maximal margin
deliver the best
generalization ability.
margin
w
Orientation of the
decision boundary
SVM—Linear Separable
Objective:
maximize the margin
maximize
2
w
wTx+b=1
wTx+b=0
wTx+b=-1
minimize
w
2
2
SVM—Linear Separable
Objective:
maximize the margin
maximize
2
w
wTx+b=1
wTx+b=0
wTx+b=-1
Support Vectors
minimize
w
2
2
SVM—Linear Separable
1 T
min
w w
w ,b
2
subject to
yi (w xi  b)  1 i  1,2,...,l
T
(P1)
The Lagrangian trick
min x
2
x
s. t. x  b
Moving the constraint to objective function
Lagrangian:
min LP  x -  ( x-b)
2
The Lagrangian trick
min LP  x -  ( x-b)
2
Optimality conditons:
LP
 2x    0
x
 0
xb
 ( x-b)  0
The Lagrangian trick
min LP  x -  ( x-b)
2
Replace
xwith  2
Solving:
max LD 

s. t .   0
2
4
 b 
SVM—Linear Separable
1 T
min
w w
w ,b
2
subject to
yi (w xi  b)  1 i  1,2,...,l
T
(P1)
SVM—Linear Separable
Lagrangian:
l
1 T
LP  w w   i [ yi (wT xi  b)  1]
2
i 1
Optimality conditons:
l
l
LP
 w   i yi xi  0  w   i yi xi
w
i 1
i 1
l
LP
  i yi  0
b
i 1
Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j (x i  x j )
α
2 i 1 j 1
i 1
Subject to
(P 2)
l
 y
i 1
i
i
i  0
0
i  1,2,...,l
Linearly Non-separable Case(Soft Margin
Optimal Hyperplane)
G54DMT - Data Mining Techniques and Applications
Linearly Non-separable Case(Soft Margin
Optimal Hyperplane)
l
1 T
min
w w  C  i
2
i 1
Subject t o
(P 3)
yi ( w T x i  b )  1   i
i  0
i  1,2 ,...l
G54DMT - Data Mining Techniques and Applications
Lagrangian
l
l
l
1 T
LP  w w  C  i   i [ yi (wxi  b)  1  i ]    ii
2
i 1
i 1
i 1
G54DMT - Data Mining Techniques and Applications
Lagrangian
l
l
LP
 w    i yi xi  0  w    i yi x i
w
i 1
i 1
l
LP
   i yi  0
b
i 1
LP
 0  C   i   i  0, i  1,...,l
 i
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Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j (x i  x j )
α
2 i 1 j 1
i 1
Subject to
(P 2)
l
 y
i 1
i
i
0
0  i  C
i  1,2 ,...,l
Problems with linear SVM
What if the decison function is not a linear?
Problems with linear SVM

Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j (xTi x j )
α
2 i 1 j 1
i 1
Subject to
(P 2)
l
 y
i 1
i
i
i  0
0
i  1,2,...,l
Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j ( (x i )T  (x j ))
α
2 i 1 j 1
i 1
Subject to
(P 2)
l
 y
i 1
i
i
i  0
0
i  1,2,...,l
Kernel Functions
A kernel function K enables the explicit
mapping of input data without exact
knowledge of
K (x, y)   (x)   (y)
Gaussian radial basis function (RBF) is
one of widely-used kernel functions
K (x, y)  e

||x y||2
2 2
Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j (xTi x j )
α
2 i 1 j 1
i 1
Subject to
l
 y
i 1
i
i
i  0
0
replace the dot product
of the inputs with the
kernel function
i  1,2,...,l
(P 2)
Dual Optimization Problem
l
1 l l
max LD    i    i j yi y j K (x i , x j )
α
2 i 1 j 1
i 1
Subject to
(P 2)
l
 y
i 1
i
i
i  0
0
i  1,2,...,l
Some kernel functions
Polynomial type:
K (u, v)  (uT v)d ,
d  1,2
Polynomial type:
K (u, v)  [(uT v)  1]d ,
d  1,2
Gaussian radial basis function (RBF)
 || u  v ||2 

K (u,v)  exp 
2
2


Multi-Layer Perceptron:
K (u, v)  tanh(uT v   )
27
G54DMT - Data Mining Techniques and Applications
Two-Spiral Pattern
Given 194 training data
points on X-Y plane: 97
of class “ red circle’’ and
another 97 of class “blue
cross ’’.
Question: how to
distinguish between
these two spirals ?
G54DMT - Data Mining Techniques and Applications
What’s the challenge?
A proper learning of these 194 training
data points
A piece of cake for a variety of methods. After all, it’s just
a limited number of 194 points
Correct assignment of an arbitrary
data point on XY plane to the right
“spiral stripe”
Very challenging since there are an infinite number of
points on XY-plane, making it the touchstone of the power
of a classification algorithm
G54DMT - Data Mining Techniques and Applications
G54DMT - Data Mining Techniques and Applications
This is exactly what we want!
G54DMT - Data Mining Techniques and Applications
References
http://www.kernel-machines.org/
http://www.support-vector.net/
AN INTRODUCTION TO SUPPORT VECTOR MACHINES
(and other kernel-based learning methods)
N. Cristianini and J. Shawe-Taylor
Cambridge University Press
2000 ISBN: 0 521 78019 5
Papers by Vapnik
C.J.C. Burges: A tutorial on Support Vector Machines. Data Mining and
Knowledge Discovery 2:121-167, 1998.

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