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Linear Discrimination and Support Vector Machines

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CHAPTER 10: Linear Discrimination Eick/Aldaydin: Topic13
Likelihood- vs. Discriminant-based Classification ,[object Object],[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Linear Discriminant ,[object Object],[object Object],[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Generalized Linear Model ,[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Two Classes Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Geometry Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Support Vector Machines ,[object Object]
Support Vector Machines ,[object Object]
Support Vector Machines ,[object Object]
Support Vector Machines ,[object Object],[object Object]
Support Vector Machines ,[object Object]
Support Vector Machines Examples are: (x1,..,xn,y) with y  {-1,1}
Support Vector Machines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Support Vector Machines ,[object Object]
Linear SVM for Non-linearly Separable Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],Measures prediction error Inverse size of margin between hyperplanes Parameter Slack variable allows constraint violation to a certain degree
Nonlinear Support Vector Machines ,[object Object],Alternative 1: Use technique that Employs non-linear decision boundaries Non-linear function
Nonlinear Support Vector Machines ,[object Object],[object Object],Alternative 2: Transform into a higher dimensional attribute space and find linear decision boundaries in this space
Nonlinear Support Vector Machines ,[object Object],[object Object],[object Object],Find a good hyperplane in the transformed space
Example: Polynomial Kernel Function ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Kernel function trick : perform computations in the original space, although we solve an optimization problem in the transformed space  more efficient; More details  Topic14.
Summary Support Vector Machines ,[object Object],[object Object],[object Object],[object Object],[object Object]
Useful Support Vector Machine Links Lecture notes are much more helpful to understand the basic ideas: http://www.ics.uci.edu/~welling/teaching/KernelsICS273B/Kernels.html   http://cerium.raunvis.hi.is/~tpr/courseware/svm/kraekjur.html   Some tools are often used in publications                  livsvm:     http://www.csie.ntu.edu.tw/~cjlin/libsvm/                   spider:     http://www.kyb.tuebingen.mpg.de/bs/people/spider/index.html   Tutorial Slides:      http://www.support-vector.net/icml-tutorial.pdf   Surveys:                 http://www.svms.org/survey/Camp00.pdf   More General Material: http://www.learning-with-kernels.org/ http://www.kernel-machines.org/ http://kernel-machines.org/publications.html http://www.support-vector.net/tutorial.html           Remarks : Thanks to Chaofan Sun for providing these links!
Optimal Separating Hyperplane Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) (Cortes and Vapnik, 1995; Vapnik, 1995) Alpaydin transparencies on Support Vector Machines —not used in lecture!
Margin ,[object Object],[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) Most α t are 0 and only a small number have α t >0; they are the support vectors
Soft Margin Hyperplane ,[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Kernel Machines ,[object Object],[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
Kernel Functions ,[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) (Cherkassky and Mulier, 1998)
SVM for Regression ,[object Object],[object Object],[object Object],Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)

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Linear Discrimination and Support Vector Machines

  • 1. CHAPTER 10: Linear Discrimination Eick/Aldaydin: Topic13
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  • 5. Two Classes Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
  • 6. Geometry Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
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  • 12. Support Vector Machines Examples are: (x1,..,xn,y) with y  {-1,1}
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  • 21. Useful Support Vector Machine Links Lecture notes are much more helpful to understand the basic ideas: http://www.ics.uci.edu/~welling/teaching/KernelsICS273B/Kernels.html   http://cerium.raunvis.hi.is/~tpr/courseware/svm/kraekjur.html   Some tools are often used in publications                  livsvm:     http://www.csie.ntu.edu.tw/~cjlin/libsvm/                   spider:     http://www.kyb.tuebingen.mpg.de/bs/people/spider/index.html   Tutorial Slides:      http://www.support-vector.net/icml-tutorial.pdf   Surveys:                 http://www.svms.org/survey/Camp00.pdf   More General Material: http://www.learning-with-kernels.org/ http://www.kernel-machines.org/ http://kernel-machines.org/publications.html http://www.support-vector.net/tutorial.html           Remarks : Thanks to Chaofan Sun for providing these links!
  • 22. Optimal Separating Hyperplane Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) (Cortes and Vapnik, 1995; Vapnik, 1995) Alpaydin transparencies on Support Vector Machines —not used in lecture!
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  • 24. Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
  • 25. Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
  • 26. Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) Most α t are 0 and only a small number have α t >0; they are the support vectors
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