super vector machines algorithms using deep

Linear Separators
• Binary classification can be viewed as the task of
separating classes in feature space:
wTx + b = 0
wTx + b < 0
wTx + b > 0
f(x) = sign(wTx + b)

Linear Separators
• Which of the linear separators is optimal?

What is a good Decision Boundary?
• Many decision
boundaries!
– The Perceptron algorithm
can be used to find such a
boundary
• Are all decision
boundaries equally
good?
4
Class 1
Class 2

Examples of Bad Decision Boundaries
5
Class 1
Class 2
Class 1
Class 2

Finding the Decision Boundary
• Let {x1, ..., xn} be our data set and let yi  {1,-1} be the class
label of xi
6
Class 1
Class 2
m
y=1
y=1
y=1
y=1
y=1
y=-1
y=-1
y=-1
y=-1
y=-1
y=-1
1

 b
x
w i
T
For yi=1
1


 b
x
w i
T
For yi=-1
   
i
i
i
T
i y
x
b
x
w
y ,
,
1 



So:

Large-margin Decision Boundary
• The decision boundary should be as far away
from the data of both classes as possible
– We should maximize the margin, m
7
Class 1
Class 2
m

• The decision boundary should classify all points correctly 
• The decision boundary can be found by solving the
following constrained optimization problem
• This is a constrained optimization problem. Solving it
requires to use Lagrange multipliers
8

• The Lagrangian is
– ai≥0
– Note that ||w||2 = wTw
9

• Setting the gradient of w.r.t. w and b to
zero, we have
10
Gradient with respect to w and b














0
,
0
b
L
k
w
L
k
 
 
 


 

























n
i
m
k
k
i
k
i
i
m
k
k
k
n
i
i
T
i
i
T
b
x
w
y
w
w
b
x
w
y
w
w
L
1 1
1
1
1
2
1
1
2
1
a
a
n: no of examples, m: dimension of the space

The Dual Problem
• If we substitute to , we have
Since
• This is a function of ai only
11

The Dual Problem
• The new objective function is in terms of ai only
• It is known as the dual problem: if we know w, we know all ai; if we know
all ai, we know w
• The original problem is known as the primal problem
• The objective function of the dual problem needs to be maximized (comes
out from the KKT theory)
• The dual problem is therefore:
12
Properties of ai when we introduce
the Lagrange multipliers
The result when we differentiate the
original Lagrangian w.r.t. b

The Dual Problem
• This is a quadratic programming (QP) problem
– A global maximum of ai can always be found
• w can be recovered by
13

Characteristics of the Solution
• Many of the ai are zero
– w is a linear combination of a small number of data
points
– This “sparse” representation can be viewed as data
compression as in the construction of knn classifier
• xi with non-zero ai are called support vectors (SV)
– The decision boundary is determined only by the SV
– Let tj (j=1, ..., s) be the indices of the s support
vectors. We can write
– Note: w need not be formed explicitly
14

A Geometrical Interpretation
15
a6=1.4
Class 1
Class 2
a1=0.8
a2=0
a3=0
a4=0
a5=0
a7=0
a8=0.6
a9=0
a10=0

Characteristics of the Solution
• For testing with a new data z
– Compute
and classify z as class 1 if the sum is positive, and
class 2 otherwise
– Note: w need not be formed explicitly
16

The Quadratic Programming Problem
• Many approaches have been proposed
– Loqo, cplex, etc. (see http://www.numerical.rl.ac.uk/qp/qp.html)
• Most are “interior-point” methods
– Start with an initial solution that can violate the constraints
– Improve this solution by optimizing the objective function
and/or reducing the amount of constraint violation
• For SVM, sequential minimal optimization (SMO) seems to
be the most popular
– A QP with two variables is trivial to solve
– Each iteration of SMO picks a pair of (ai,aj) and solve the QP
with these two variables; repeat until convergence
• In practice, we can just regard the QP solver as a “black-
box” without bothering how it works
17

Non-linearly Separable Problems
• We allow “error” xi in classification; it is based on the output
of the discriminant function wTx+b
• xi approximates the number of misclassified samples
18
Class 1
Class 2

Soft Margin Hyperplane
• The new conditions become
– xi are “slack variables” in optimization
– Note that xi=0 if there is no error for xi
– xi is an upper bound of the number of errors
• We want to minimize
• C : tradeoff parameter between error and margin
19



n
i
i
C
w
1
2
2
1
x

The Optimization Problem
20
 
  

 









n
i
i
i
n
i
i
T
i
i
i
n
i
i
T
b
x
w
y
C
w
w
L
1
1
1
1
2
1
x

x
a
x
0
1







n
i
ij
i
i
j
j
x
y
w
w
L
a 0
1

 

n
i
i
i
i x
y
w


a
0






j
j
j
C
L

a
x
0
1






n
i
i
i
y
b
L
a
With α and μ Lagrange multipliers, POSITIVE

The Dual Problem

 
 



n
i
i
j
T
i
j
i
n
i
n
j
j
i x
x
y
y
L
1
1 1
2
1
a
a
a



 



 

 
























n
i
i
i
n
i
n
j
i
T
j
j
j
i
i
i
n
i
i
j
T
i
j
i
n
i
n
j
j
i
b
x
x
y
y
C
x
x
y
y
L
1
1 1
1
1 1
1
2
1
x

a
x
a
x
a
a


j
j
C 
a 

0
1



n
i
i
i
y a
With

The Optimization Problem
• The dual of this new constrained optimization problem is
• New constrainsderive from since μ and α are
positive.
• w is recovered as
• This is very similar to the optimization problem in the linear
separable case, except that there is an upper bound C on ai
now
• Once again, a QP solver can be used to find ai
22
j
j
C 
a 


• The algorithm try to keep ξ null, maximising the
margin
• The algorithm does not minimise the number of
error. Instead, it minimises the sum of distances fron
the hyperplane
• When C increases the number of errors tend to
lower. At the limit of C tending to infinite, the
solution tend to that given by the hard margin
formulation, with 0 errors
3/11/2024 23



n
i
i
C
w
1
2
2
1
x

Extension to Non-linear Decision
Boundary
• So far, we have only considered large-margin classifier with
a linear decision boundary
• How to generalize it to become nonlinear?
• Key idea: transform xi to a higher dimensional space to
“make life easier”
– Input space: the space the point xi are located
– Feature space: the space of f(xi) after transformation
• Why transform?
– Linear operation in the feature space is equivalent to non-linear
operation in input space
– Classification can become easier with a proper transformation.
In the XOR problem, for example, adding a new feature of x1x2
make the problem linearly separable
25

XOR
X Y
0 0 0
0 1 1
1 0 1
1 1 0
26
Is not linearly separable
X Y XY
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
Is linearly separable

Transforming the Data
• Computation in the feature space can be costly
because it is high dimensional
– The feature space is typically infinite-dimensional!
• The kernel trick comes to rescue
28
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f(.)
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
Feature space
Input space
Note: feature space is of higher dimension
than the input space in practice

Transforming the Data
• Computation in the feature space can be costly
because it is high dimensional
– The feature space is typically infinite-dimensional!
• The kernel trick comes to rescue
29
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f(.)
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
f( )
Feature space
Input space
Note: feature space is of higher dimension
than the input space in practice

The Kernel Trick
• Recall the SVM optimization problem
• The data points only appear as inner product
• As long as we can calculate the inner product in the
feature space, we do not need the mapping explicitly
• Many common geometric operations (angles,
distances) can be expressed by inner products
• Define the kernel function K by
30

An Example for f(.) and K(.,.)
• Suppose f(.) is given as follows
• An inner product in the feature space is
• So, if we define the kernel function as follows, there is no
need to carry out f(.) explicitly
• This use of kernel function to avoid carrying out f(.)
explicitly is known as the kernel trick
31

Kernels
• Given a mapping:
a kernel is represented as the inner product
A kernel must satisfy the Mercer’s condition:
32
φ(x)
x 


i
i
i φ
φ
K (y)
(x)
y
x )
,
(

 


 0
)
(
)
(
)
(
0
)
(
such that
)
( 2
y
x
y
x
y
x,
x
x
x d
d
g
g
K
d
g
g

Modification Due to Kernel Function
• Change all inner products to kernel functions
• For training,
33
Original
With kernel
function

Modification Due to Kernel Function
• For testing, the new data z is classified as class
1 if f 0, and as class 2 if f <0
34
Original
With kernel
function

More on Kernel Functions
• Since the training of SVM only requires the value of
K(xi, xj), there is no restriction of the form of xi and xj
– xi can be a sequence or a tree, instead of a feature vector
• K(xi, xj) is just a similarity measure comparing xi and xj
• For a test object z, the discriminant function essentially
is a weighted sum of the similarity between z and a
pre-selected set of objects (the support vectors)
35

Example
• Suppose we have 5 1D data points
– x1=1, x2=2, x3=4, x4=5, x5=6, with 1, 2, 6 as class 1
and 4, 5 as class 2  y1=1, y2=1, y3=-1, y4=-1, y5=1
36

Example
37
1 2 4 5 6
class 2 class 1
class 1

Example
• We use the polynomial kernel of degree 2
– K(x,y) = (xy+1)2
– C is set to 100
• We first find ai (i=1, …, 5) by
38

Example
• By using a QP solver, we get
– a1=0, a2=2.5, a3=0, a4=7.333, a5=4.833
– Note that the constraints are indeed satisfied
– The support vectors are {x2=2, x4=5, x5=6}
• The discriminant function is
• b is recovered by solving f(2)=1 or by f(5)=-1 or by f(6)=1,
• All three give b=9
39

Example
40
Value of discriminant function
1 2 4 5 6
class 2 class 1
class 1

Kernel Functions
• In practical use of SVM, the user specifies the kernel
function; the transformation f(.) is not explicitly stated
• Given a kernel function K(xi, xj), the transformation f(.)
is given by its eigenfunctions (a concept in functional
analysis)
– Eigenfunctions can be difficult to construct explicitly
– This is why people only specify the kernel function without
worrying about the exact transformation
• Another view: kernel function, being an inner product,
is really a similarity measure between the objects
41

A kernel is associated to a
transformation
– Given a kernel, in principle it should be recovered the
transformation in the feature space that originates it.
– K(x,y) = (xy+1)2= x2y2+2xy+1
It corresponds the transformation
3/11/2024 42











1
2
2
x
x
x

Examples of Kernel Functions
• Polynomial kernel up to degree d
• Polynomial kernel up to degree d
• Radial basis function kernel with width s
– The feature space is infinite-dimensional
• Sigmoid with parameter k and q
– It does not satisfy the Mercer condition on all k and q
43

Building new kernels
• If k1(x,y) and k2(x,y) are two valid kernels then the
following kernels are valid
– Linear Combination
– Exponential
– Product
– Polymomial tranfsormation (Q: polymonial with non
negative coeffients)
– Function product (f: any function)
45
)
,
(
)
,
(
)
,
( 2
2
1
1 y
x
k
c
y
x
k
c
y
x
k 

 
)
,
(
exp
)
,
( 1 y
x
k
y
x
k 
)
,
(
)
,
(
)
,
( 2
1 y
x
k
y
x
k
y
x
k 

 
)
,
(
)
,
( 1 y
x
k
Q
y
x
k 
)
(
)
,
(
)
(
)
,
( 1 y
f
y
x
k
x
f
y
x
k 

Ploynomial kernel
Ben-Hur et al, PLOS computational Biology 4 (2008)
46

Gaussian RBF kernel
Ben-Hur et al, PLOS computational Biology 4 (2008)
47

Spectral kernel for sequences
• Given a DNA sequence x we can count the
number of bases (4-D feature space)
• Or the number of dimers (16-D space)
• Or l-mers (4l –D space)
• The spectral kernel is
3/11/2024 48
)
,
,
,
(
)
(
1 T
G
C
A n
n
n
n
x 
f
,..)
,
,
,
,
,
,
,
(
)
(
2 CT
CG
CC
CA
AT
AG
AC
AA n
n
n
n
n
n
n
n
x 
f
   
y
x
y
x
k l
l
l f
f 

)
,
(

Choosing the Kernel Function
• Probably the most tricky part of using SVM.
• The kernel function is important because it creates the
kernel matrix, which summarizes all the data
• Many principles have been proposed (diffusion kernel,
Fisher kernel, string kernel, …)
• There is even research to estimate the kernel matrix from
available information
• In practice, a low degree polynomial kernel or RBF kernel
with a reasonable width is a good initial try
• Note that SVM with RBF kernel is closely related to RBF
neural networks, with the centers of the radial basis
functions automatically chosen for SVM
49

Other Aspects of SVM
• How to use SVM for multi-class classification?
– One can change the QP formulation to become multi-class
– More often, multiple binary classifiers are combined
• See DHS 5.2.2 for some discussion
– One can train multiple one-versus-all classifiers, or
combine multiple pairwise classifiers “intelligently”
• How to interpret the SVM discriminant function value
as probability?
– By performing logistic regression on the SVM output of a
set of data (validation set) that is not used for training
• Some SVM software (like libsvm) have these features
built-in
50

Active Support Vector Learning
P. Mitra, B. Uma Shankar and S. K. Pal, Segmentation of multispectral remote sensing
Images using active support vector machines, Pattern Recognition Letters, 2004.

Software
• A list of SVM implementation can be found at
http://www.kernel-
machines.org/software.html
• Some implementation (such as LIBSVM) can
handle multi-class classification
• SVMLight is among one of the earliest
implementation of SVM
• Several Matlab toolboxes for SVM are also
available
53

Summary: Steps for Classification
• Prepare the pattern matrix
• Select the kernel function to use
• Select the parameter of the kernel function and
the value of C
– You can use the values suggested by the SVM
software, or you can set apart a validation set to
determine the values of the parameter
• Execute the training algorithm and obtain the ai
• Unseen data can be classified using the ai and the
support vectors
54

Strengths and Weaknesses of SVM
• Strengths
– Training is relatively easy
• No local optimal, unlike in neural networks
– It scales relatively well to high dimensional data
– Tradeoff between classifier complexity and error can
be controlled explicitly
– Non-traditional data like strings and trees can be used
as input to SVM, instead of feature vectors
• Weaknesses
– Need to choose a “good” kernel function.
55

Conclusion
• SVM is a useful alternative to neural networks
• Two key concepts of SVM: maximize the
margin and the kernel trick
• Many SVM implementations are available on
the web for you to try on your data set!
56

Resources
• http://www.kernel-machines.org/
• http://www.support-vector.net/
• http://www.support-vector.net/icml-
tutorial.pdf
• http://www.kernel-
machines.org/papers/tutorial-nips.ps.gz
• http://www.clopinet.com/isabelle/Projects/SV
M/applist.html
57

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