Lecture slides on SVM as a part of a course on Neural Networks.

1. CHAPTER 06
SUPPORT VECTOR MACHINES
CSC445: Neural Networks
Prof. Dr. Mostafa Gadal-Haqq M. Mostafa
Computer Science Department
Faculty of Computer & Information Sciences
AIN SHAMS UNIVERSITY
(some of the figures in this presentation are copyrighted to Pearson Education, Inc.)

2. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
 Introduction
 Optimal Hyperplane for Linearly Separable Pattern
 Quadratic Optimization for Finding the Optimal Hyperplan
 Optimal Hyperplane for Nonseparable Patterns
 Underlying Philosophy of SVM for Pattern Calssification
 SVM viewed as Kernel Machine
 The XOR problem
 Computer Experiment
2
Outlines

3. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq 3
Introduction
 The main idea of the SVMs may be summed up as
follows:
 “Given a training samples, the SVM constructs a
hyperplane as decision surface in such a way the
margin of separation between positive and negative
examples is maximized.”

4. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq 4
Linearly Separable Patterns
 SVM is a binary learning machine.
 Binary classification is the task of separating classes in
feature space.
wTx + b = 0
wTx + b < 0
wTx + b > 0
bxwxg T


)(

5. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq 5
Linearly Separable Patterns
 Which of the linear separators is optimal?

6. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Optimal Decision Boundary
 The optimal decision boundary is the one that
maximize the margin 
6
r
ρ

7. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The Margin 
7
|||| w
w
rxx P 




8. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The Margin 
||||)(then
,0since
||||
)()(
||||
,)(
wrxg
bxw
w
w
w
rbxwxg
w
w
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P
T
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P
T
P
T

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

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8

9. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The Margin 










1
||||
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)(
11)(
dif
w
dif
w
w
xg
r
dforbxwxg T

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

9
r
ρ
1bxwT 
1 bxwT 
0 bxwT 
||||
2
2
w
r 
Then the margin is given as:

10. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Optimal Decision Boundary
 Let {x1, ..., xn} be our data set and let di  {1,-1} be the
class label of xi
 The decision boundary should classify all points
correctly.
 That is, we have a constrained optimization problem
Maximize  = 𝟐𝒓 =
𝟐
𝒘
, or Minimize 𝒘
Subject to 𝒅𝒊(𝒘 𝑻 𝒙 ± 𝒃) ≥ 𝟏
10

11. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The Optimization Problem
 Introduce Lagrange multipliers ,
 That is, the Lagrange function:
Is to be minimized with respect to w and b, i.e,
𝜕𝑱(𝒘,𝒃,)
𝜕𝒘
= 𝟎 ; and
𝜕𝑱(𝒘,𝒃, )
𝜕𝒃
= 𝟎
)1][(||||
2
1
),,(
1
2
 
bxwdwbwJ i
T
i
N
i
i
11

12. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Solving the Optimization Problem
 Need to optimize a quadratic function subject to linear
constraints.
 The solution involves constructing a dual problem where a
Lagrange multiplier αi is associated with every constraint in the
primary problem:
Find 𝛼1…𝛼 𝑁such that
𝑸 𝜶 = 𝛼𝑖 −
1
2
𝛼𝑖 𝛼𝑗 𝑑𝑖 𝑑𝑗x 𝑖x𝑗𝑗𝑖
𝑵
𝒊=𝟏
is maximized and
(1) 𝛼𝑖 𝑑𝑖𝑗
(2) 𝛼1 ≥ 0 ∀ 𝑖
12

13. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The Optimization Problem
 The solution has the form:
and such that 𝒊 ≠ 𝟎
 Each non-zero αi indicates that corresponding xi is a support vector.
 Then the classifying function will have the form:
 Notice that it relies on an inner product between the test point x and the
support vectors xi
 Also keep in mind that solving the optimization problem involved computing
the inner products xi
Txj between all training points!
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ii
N
i
i xd

1
w  iii
N
i
idb xx1
1

 
bdxg iii
N
i
i  
xx)(
1


14. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
6=1.4
The Optimization Problem
 Support vectors are samples that have non-zero 
Class 1
Class 2
1=0.8
2=0
3=0
4=0
5=0
7=0
8=0.6
9=0
10=0

15. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Optimal Hyperplane for Nonseparable Patterns
Figure 6.3 Soft margin hyperplane (a) Data point xi (belonging to class C1,
represented by a small square) falls inside the region of separation, but on the
correct side of the decision surface. (b) Data point xi (belonging to class C2,
represented by a small circle) falls on the wrong side of the decision surface.
15

16. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Optimal Hyperplane for Nonseparable Patterns
 We allow “error” xi in classification
16
ξi
ξi

17. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Soft Margin Hyperplane
 The old formulation:
 The new formulation incorporating relaxed variables:
Parameter C can be viewed as a way to control overfitting.
17
Find w and b such that
∅ 𝑾 =
𝟏
𝟐
𝑾 𝑻
𝑾 is minimized and for all {(xi ,yi)}
Subject to: 𝒅𝒊(𝒘 𝑻
𝒙 ± 𝒃) ≥ 𝟏
Find w and b such that
∅ 𝐖 =
𝟏
𝟐
𝐖 𝐓 𝐖 + 𝐜 𝝃𝒊𝒊 is minimized for all {(xi ,yi)}
Subject to: 𝒅𝒊(𝒘 𝑻 𝒙 ± 𝒃) ≥ 𝟏 , and ξi ≥ 0 for all i

18. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Soft Margin Hyperplane
 Again, xi with non-zero αi will be support vectors.
 Solution to the dual problem is:
𝑾 = 𝜶𝒊 𝒅𝒊 𝒙𝒊𝒊
and
𝒃 = 𝒅𝒊 𝟏 − 𝝃𝒊 − 𝑾 𝑻
𝒙𝒊
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19. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Extension to Non-linear Decision Boundary
 Key idea: transform xi to a higher dimensional space
 Input space: the space of xi
 Feature space: the “kernel” space of f(xi)
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f( )
f( )
f( )
f( )f( )
f( )
f( )
f( )
f(.)
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
Feature spaceInput space

20. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Kernel Trick
 The linear classifier relies on inner product between
vectors:
𝑲 𝐱 𝒊, 𝐱 𝒋 = 𝐱𝒊
𝑻 𝐱 𝒋
If every datapoint is mapped into high-dimensional space
via some transformation Φ: x → φ(x), the inner product
becomes:
𝑲 𝐱 𝒊, 𝐱 𝒋 = 𝛟 𝐱𝐢
𝑻 𝛟(𝐱 𝒋)
 A kernel function is some function that corresponds to
an inner product into some feature space.
 K (x, xj) needs to satisfy a technical condition (Mercer
condition) in order for f(.) to exist
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21. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Mercer’s Theorem
 𝑲 = 𝒌(𝒙𝒊, 𝒙𝒋) ∀𝒊, 𝒋 has to be non-negative definite or
positive semidefinite , that is, it satisfies:
𝒂 𝑻K𝒂 ≥ 𝟎
 Some of kernel functions that satisfy Mercer’s condition:
21

22. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The SVM viewed as Kernel Machine
Figure 6.5 Architecture of support vector machine, using a
radial-basis function network.
22

23. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The XOR Problem
 For the two dimensional vectors x=[x1 x2];
 Define the following Kernel:
𝒌 x,x𝒊 = 𝟏 + x 𝑻
x𝒊
2
 Need to show that
K(xi,xj)= φ(xi)Tφ(xj)
K(xi,xj)=(1 + xi
Txj)2
= 1+ xi1
2xj1
2 + 2 xi1xj1 xi2xj2+ xi2
2xj2
2 + 2xi1xj1 + 2xi2xj2=
= [1 xi1
2 √2 xi1xi2 xi2
2 √2xi1 √2xi2]T [1 xj1
2 √2 xj1xj2 xj2
2 √2xj1 √2xj2]
= φ(xi)Tφ(xj),
where
φ(x) = [1 x1
2 √2 x1x2 x2
2 √2x1 √2x2]
23

24. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
The XOR Problem
 Which give the optimal hyperplane as:
−𝒙 𝟏 𝒙 𝟐 = 𝟎
 This yields
Figure 6.6 (a) Polynomial machine for solving the XOR problem. (b) Induced
images in the feature space due to the four data points of the XOR problem.
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(1, -1)
(-1,1)
(-1, -1)
(1,1)
-1.0

25. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Conclusion
 SVM is a useful alternative to neural networks
 Two key concepts of SVM: maximize the margin
and the kernel trick
 Many active research is taking place on areas
related to SVM
 Many SVM implementations are available on the
web for you to try on your data set!
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26. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Computer Experiment
Figure 6.7 Experiment on SVM for the double-moon of Fig. 1.8 with
distance d = –6.
26

27. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
Computer Experiment
Figure 6.8 Experiment on SVM for the double-moon of Fig. 1.8 with
distance d = –6.5.
27

28. Principal Component
Analysis (PCA)
Next Time
28