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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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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

  • possible solution: support vector machines (SVM)


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


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.


Maximum Margin Classifiers (3)


  • 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


Maximum Margin Classifiers (4)


7.1.1 Overlapping class distributions


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


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.


7.1.2 Relation to logistic regression


7.1.3 Multi-class SVM


7.1.4 SVMs for regression


7.1.4 SVMs for regression


7.1.4 SVMs for regression


7.1.4 SVMs for regression


7.1.4 SVMs for regression &7.1.5 Computational learning theory


Relevance Vector Machines (1)


Relevance Vector Machines (2)7.2.1 RVM for regression


Relevance Vector Machines (3)7.2.1 RVM for regression


Relevance Vector Machines (4)7.2.1 RVM for regression


Relevance Vector Machines (5)7.2.1 RVM for regression


Relevance Vector Machines (6)7.2.1 RVM for regression


Relevance Vector Machines (8)7.2.2 Analysis of sparsity


Relevance Vector Machines (9)7.2.3 RVM for classification


Relevance Vector Machines (10)7.2.3 RVM for classification


Relevance Vector Machines (11)7.2.3 RVM for classification


Relevance Vector Machines (12)7.2.3 RVM for classification


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