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Support Vector Machines Part 2

Support Vector Machines Part 2. Recap of SVM algorithm. Given training set S = {( x 1 , y 1 ), ( x 2 , y 2 ), ..., ( x m , y m ) | ( x i , y i )  n {+1, -1} Choose a cheap-to-compute kernel function k ( x , z )

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Support Vector Machines Part 2

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  1. Support Vector MachinesPart 2

  2. Recap of SVM algorithm Given training set S = {(x1, y1), (x2, y2), ..., (xm, ym) | (xi, yi)n{+1, -1} • Choose a cheap-to-compute kernel function k(x,z) 2. Apply quadratic programming procedure (using the kernel function k) to find {i}and bias b, where i≠ 0 only if xiis a support vector.

  3. 3. Now, given a new instance, x, find the classification of x by computing

  4. Clarifications from last time

  5. Length of the margin margin http://nlp.stanford.edu/IR-book/html/htmledition/img1260.png Without changing the problem, we can rescale our data to set a = 1

  6. w is perpendicular to decision boundary

  7. More on Kernels • So far we’ve seen kernels that map instances in nto instances in z where z > n. • One way to create a kernel: Figure out appropriate feature space Φ(x), and find kernel function k which defines inner product on that space. • More practically, we usually don’t know appropriate feature space Φ(x). • What people do in practice is either: • Use one of the “classic” kernels (e.g., polynomial), or • Define their own function that is appropriate for their task, and show that it qualifies as a kernel.

  8. How to define your own kernel • Given training data (x1, x2, ..., xn) • Algorithm for SVM learning uses kernel matrix (also called Gram matrix): • We can choose some function k, and compute the kernel matrix K using the training data. • We just have to guarantee that our kernel defines an inner product on some feature space. • Not as hard as it sounds.

  9. What counts as a kernel? • Mercer’s Theorem: If the kernel matrix K is “symmetric positive definite”, it defines a kernel on the training data, that is, it defines an inner product in some feature space. • We don’t even have to know what that feature space is! It can have a huge number of dimensions.

  10. In-class exercises Note for part (c):

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