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VC theory, Support vectors and Hedged prediction technology

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Overfitting in classification Assume a family C of classifiers of points in feature space F. A family of classifiers is a map from C F to {0,1} (Negative and positive class). For each subset X of F and each c in C, c(X) defines a partitioning of X into two classes. C shatters X if every partitioning of X is accomplished by some c in C If every point set X of size d is shattered by C, then the VC dimension is at least d. If a point set of d+1 elements cannot be shattered by C, then the VC-dimension is at most d.

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VC-dimension of hyperplanes The set of points on the line shatters any two points, but not three The set of lines in the plane shatters any three non-collinear points, but no four points. Any d+2 points in E^d can be partitioned into two blocks whose convex hulls intersect. VC-dimension of hyperplanes in E^d is thus d+1.

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Why VC-dimension? Elegant and pedagogical, not very useful. Bounds future error of classifier, PAC-learning. Exchangeable distribution of (xi, yi). For first N points, training error for c is observed error rate for c. Goodness of selecting from C a classifier with best performance on training set depends on VC-dimension h:

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Why VC-dimension?

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Classify with hyperplanes Frank Rosenblatt (1928 – 1971) Pioneering work in classifying by hyperplanes in high-dimensional spaces. Criticized by Minsky-Papert, since real classes are not normally linearly separable. ANN research taken up again in 1980:s, with non-linear mappings to get improved separation. Predecessor to SVM/kernel methods

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Find parallel hyperplanes Separate examples by wide margin hyperplanes (classifications). Enclose examples between hyperplanes (regression). If necessary, non-linearly map examples to high-dimensional space where they are better separated.

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Find parallel hyperplanes Classification Red true separating plane. Blue: wide margin separation in sample Classify by plane between blue planes

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Find parallel hyperplanes Regression Red: true central plane. Blue: narrowest margin enclosing sample New xk : predict yk so (xk, yk) lies on mid- plane (dotted).

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From vector to scalar product

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Soft Margins

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Quadratic programming goes through also with soft margins. Specification of softness constant C is part of most packages. However, no prior rule for setting C is established, and experimentation is necessary for each application. Choice is between narrowing margin, allowing more outliers, and using a more liberal kernel (to be described).

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SVM packages Inputs xi, yi, and KERNEL and SOFTNESS information Only output is , non-zero coefficients indicate support vectors. Hyperplane obtained by

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Kernel Trick

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Example: 2D space (x1,x2). Map to 5D space (c1*x1, c2*x2, c3*x1^2, c4*x1*x2, c5*x2^2). K(x,y)=(x y+1)^2 =2*x1*y1+2*x2*y2+x1^2*y1^2+x2^2*y2^2+2*x1*x2*y1*y2+1 = (x) (y), Where (x)= ((x1,x2)) = (√2x1, √2x2, x1^2, √2x1*x2, x2^2). Hyperplanes in R^5 are mapped back to conic sections in R^2!!

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Kernel Trick Gaussian Kernel: K(x,y) = exp(-||x-y||^2/ 2

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