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Sparse Kernel Machines

Sparse Kernel Machines. Introduction. Kernel function k(xn,xm) must be evaluated for all pairs of training points Advantageous (computationally) to look at kernel algorithms that have sparse solutions new predictions depend only on kernel function evaluated at a few training points

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Sparse Kernel Machines

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  1. Sparse Kernel Machines

  2. Introduction • Kernel function k(xn,xm) must be evaluated for all pairs of training points • Advantageous (computationally) to look at kernel algorithms that have sparse solutions • new predictions depend only on kernel function evaluated at a few training points • possible solution: support vector machines (SVM)

  3. Maximum Margin Classifiers (1) • Setup: • y(x) = w’φ(x) + b, φ(x) = feature-space transformation Training data comprises • N input data vectors x1,...,xN • N target vectors t1,...,tNtn∈{-1,1} • new samples classified according to sign of y(x) • assume linearly separable data for the moment • y(x)>0 for points of class +1 (i.e. tn = +1) • y(x)<0 for points having tn = -1 • therefore,tny(xn) > 0 for all training points

  4. Maximum Margin Classifiers (2) • There may be many solutions to separate the data SVM tries to minimize the generalization error by introducing the concept of a margin i.e. smallest distance between the decision boundary and any of the samples • Decision boundary is chosen to be that for which the margin is maximized • decision boundary thus only dependent on points closest to it.

  5. Maximum Margin Classifiers (3)

  6. If we consider only points that are correctly classified, then we have tny(xn) > 0 for all n • The distance of xn to the decision surface is then • tny(xn) • w = tn(wTφ(xn) + b) • w • ● Optimize w and b to maximize this distance => solve • (7.2) • (7.3) • ● Quite hard to solve => need to get around it

  7. Maximum Margin Classifiers (4)

  8. 7.1.1 Overlapping class distributions

  9. An alternative, equivalent formulation of the support vector machine, known as the ν-SVM

  10. the parameter ν, which replaces C, can beinterpreted as both an upper bound on the fraction of margin errors (points for whichξn > 0 and hence which lie on the wrong side of the margin boundary and which mayor may not be misclassified) and a lower bound on the fraction of support vectors.

  11. 7.1.2 Relation to logistic regression

  12. 7.1.3 Multi-class SVM

  13. 7.1.4 SVMs for regression

  14. 7.1.4 SVMs for regression

  15. 7.1.4 SVMs for regression

  16. 7.1.4 SVMs for regression

  17. 7.1.4 SVMs for regression &7.1.5 Computational learning theory

  18. Relevance Vector Machines (1)

  19. Relevance Vector Machines (2)7.2.1 RVM for regression

  20. Relevance Vector Machines (3)7.2.1 RVM for regression

  21. Relevance Vector Machines (4)7.2.1 RVM for regression

  22. Relevance Vector Machines (5)7.2.1 RVM for regression

  23. Relevance Vector Machines (6)7.2.1 RVM for regression

  24. Relevance Vector Machines (8)7.2.2 Analysis of sparsity

  25. Relevance Vector Machines (9)7.2.3 RVM for classification

  26. Relevance Vector Machines (10)7.2.3 RVM for classification

  27. Relevance Vector Machines (11)7.2.3 RVM for classification

  28. Relevance Vector Machines (12)7.2.3 RVM for classification

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