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Support Vector Machines (part 1)

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  1. Support Vector Machines (part 1) Face Recognition & Biometric Systsems

  2. Plan of the lecture • Problem of classification • SVM for solving linear problems • training • classification • Application of convolution kernels Face Recognition & Biometric Systsems

  3. Bibliography Corrina Cortez, Vladimir Vapnik Support-Vector Networks Face Recognition & Biometric Systsems

  4. Classification problem • Aim: classification of an element to one of defined classes • Two stages: • training • classification of samples • Available solutions: • Artificial Neural Networks • Support Vector Machines • other classifiers Face Recognition & Biometric Systsems

  5. Classification problem • Training set - requirements: • classified • representative • Training process: • aims at finding general rules • a risk of overfitting to the training set (especially when it is not representative) Face Recognition & Biometric Systsems

  6. Classification problem • Classification of samples: • must be preceded by the training stage • applies rules derived from the training • Number of classes: • SVM solves two-class problems • it is possible to solve multi-class problems basing on two-class problems Face Recognition & Biometric Systsems

  7. Classification problem • Linearly separable Face Recognition & Biometric Systsems

  8. Classification problem • Non-linearly separable Face Recognition & Biometric Systsems

  9. Classification problem • Training with error (soft margin) Face Recognition & Biometric Systsems

  10. Classification problem • Margin maximisation Face Recognition & Biometric Systsems

  11. Linear separability • Data set: (y1,x1),...,(yl,xl), yi{-1,1} • Vector w, scalar value b: w•xi + b  1 for yi = 1 w•xi + b  -1 for yi = -1 hence yi (w•xi + b)  1 • The condition must be fulfilled for the whole data set Face Recognition & Biometric Systsems

  12. SVM – training • SVM solves linear separable two-class problems • other cases transformed to the basic problem • Optimal hyperplane • margin between samples of two classes • margin maximisation Face Recognition & Biometric Systsems

  13. SVM – training • Optimal hyperplane: w0 •x + b0 = 0 • 2D example – hyperplane is a line • Margin width (without b): Face Recognition & Biometric Systsems

  14. SVM – training • Optimal width: • Maximisation of , minimisation of w0 •w0 • Limitation: yi (w•xi + b)  1 Face Recognition & Biometric Systsems

  15. SVM – training • Margin: • Optimal hyperplane: • yi – class identifier • i – Lagrange multipliers • A problem: how to find i? Face Recognition & Biometric Systsems

  16. SVM – training • Function maximisation: 1 – unitary vector (l – dimensional) D – l x l matrix: Face Recognition & Biometric Systsems

  17. SVM – training • Optimisation limits: • Optimisation based on the gradient method Face Recognition & Biometric Systsems

  18. SVM – training • Lagrange multipliers : • non-zero values for support vectors • equal zero for other vectors (majority) • Training set after the training: • support vectors (a small subset of the training set) •  coefficients for every vector Face Recognition & Biometric Systsems

  19. SVM – classification • Calculate y for a vector which is to be classified: xr, xs – support vectors from both classes • Classification decision Face Recognition & Biometric Systsems

  20. SVM – limitations • SVM conditions: • solves two-class problem • linear separability of data • A XOR problem: Face Recognition & Biometric Systsems

  21. SVM – limitations • Possibilities of enhancement: • SVM for non-linear data – too complicated calculations • transformation of the data, so that they are linearly separable • Mapping into higher dimension • example of XOR in 2D mapped into 3D Face Recognition & Biometric Systsems

  22. Convolution kernels • Function: • Mapping into higher dimension: x   (x) • Calculations use scalar product of vectors, not the vectors themselves • Kernels of convolution may be used instead of scalar products • No need to find  function Face Recognition & Biometric Systsems

  23. Convolution kernels • Training with convolution kernels Face Recognition & Biometric Systsems

  24. Convolution kernels • Classification with convolution kernels xr, xs – support vectors from both classes Face Recognition & Biometric Systsems

  25. Convolution kernels • Linear • Polynomial • RBF (radial basis functions) Face Recognition & Biometric Systsems

  26. Summary • Classifiers • Basic problem: • two-class linear separable data set • solved by the SVM • Enhancement • convolution kernels – SVM for non-linear separable data Face Recognition & Biometric Systsems

  27. Thank you for your attention! • Next week Support Vector Machines – continued... • multi-class cases • soft margin training • applications to face recognition Face Recognition & Biometric Systsems

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