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Support Vector Machines

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Advanced Computing Seminar Data Mining and Its Industrial Applications — Chapter 8 — Support Vector Machines ,[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
History ,[object Object],[object Object],[object Object],[object Object]
What is SVM? ,[object Object],[object Object],[object Object],[object Object],[object Object]
Linear Classifiers  y est denotes +1 denotes -1 How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore f x f ( x , w ,b ) = sign( w . x - b )
Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
Maximum Margin f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) The maximum margin linear classifier is the linear classifier with the maximum margin. This is the simplest kind of SVM (Called an LSVM) Linear SVM Copyright © 2001, 2003, Andrew W. Moore
Model of Linear Classification ,[object Object],[object Object],[object Object]
The concept of Hyperplane ,[object Object],[object Object]
Tuning the Hyperplane (w,b) ,[object Object],[object Object],[object Object],[object Object],[object Object]
The Perceptron Algorithm ,[object Object]
The Geometric margin -> ,[object Object]
The Geometric margin ,[object Object],[object Object],[object Object],[object Object],[object Object]
How to Find the optimal solution ？ ,[object Object],[object Object],[object Object],[object Object]
The Maximal Margin Classifier ,[object Object],[object Object],[object Object],[object Object]
Support Vector Classifiers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Formalizi the geometric margin ,[object Object],[object Object],[object Object]
Minimizing the norm -> ,[object Object],[object Object]
Minimizing the norm -> ,[object Object],[object Object],[object Object]
Minimizing the norm ,[object Object],[object Object],[object Object],[object Object]
The Support Vector ,[object Object],[object Object],[object Object],[object Object]
The optimal hypothesis (w,b) ,[object Object],[object Object]
Soft Margin Optimization ,[object Object],[object Object],[object Object]
Non-linear Classification ,[object Object],[object Object],[object Object],[object Object],[object Object]
A learning machine ,[object Object],f x  y est  is some vector of adjustable parameters
Some definitions ,[object Object],[object Object],[object Object],[object Object],Official terminology
Some definitions ,[object Object],[object Object],[object Object],[object Object],Official terminology R = #training set data points
Vapnik-Chervonenkis Dimension ,[object Object],[object Object],[object Object],This gives us a way to estimate the error on future data based only on the training error and the VC-dimension of f
Structural Risk Minimization ,[object Object],[object Object],[object Object],[object Object],[object Object],f 4 4 f 5 5 f 6 6  f 3 3 f 2 2 f 1 1 Choice Probable upper bound on TESTERR VC-Conf TRAINERR f i i
Kernel-Induced Feature Space ,[object Object]
Implicit Mapping into Feature Space ,[object Object]
Kernel Function ,[object Object],[object Object],[object Object]
Kernel function
The Polynomial Kernel ,[object Object],[object Object]
 
 
Text Categorization Inductive learning Inpute : Output : f(x) = confidence(class) In the case of text classification ,the attribute are words in the document ,and the classes are the categories.
PROPERTIES OF TEXT-CLASSIFICATION TASKS ,[object Object],[object Object],[object Object]
Text representation and feature selection ,[object Object],[object Object],[object Object],[object Object],[object Object]
 
Learning SVMS ,[object Object],[object Object],[object Object],[object Object]
Processing ,[object Object],[object Object],[object Object]
An example - Reuters ( trends & controversies) ,[object Object],[object Object],[object Object],[object Object]
 
Text Categorization Results Dumais et al. (1998)
Apply to the Linear Classifier ,[object Object],[object Object]
SVMs and PAC Learning ,[object Object],[object Object],[object Object]
PAC and the Number of Support Vectors ,[object Object],[object Object],[object Object],[object Object],[object Object]
VC-dimension of an SVM ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Copyright © 2001, 2003, Andrew W. Moore
Finding Non-Linear Separating Surfaces ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object]
References ,[object Object],[object Object],[object Object],[object Object],[object Object]
www.intsci.ac.cn/shizz / ,[object Object]

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Support Vector Machines

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  • 5. Linear Classifiers  y est denotes +1 denotes -1 How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore f x f ( x , w ,b ) = sign( w . x - b )
  • 6. Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
  • 7. Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
  • 8. Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
  • 9. Linear Classifiers f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) How would you classify this data? Copyright © 2001, 2003, Andrew W. Moore
  • 10. Maximum Margin f x  y est denotes +1 denotes -1 f ( x , w ,b ) = sign( w . x - b ) The maximum margin linear classifier is the linear classifier with the maximum margin. This is the simplest kind of SVM (Called an LSVM) Linear SVM Copyright © 2001, 2003, Andrew W. Moore
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  • 40. Text Categorization Inductive learning Inpute : Output : f(x) = confidence(class) In the case of text classification ,the attribute are words in the document ,and the classes are the categories.
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  • 48. Text Categorization Results Dumais et al. (1998)
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