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An Introduction to Support Vector Machine (SVM)
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Famous Examples that helped SVM become popular
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Classification Everyday, all the time we classify things.
Eg crossing the street: Is there a car coming? At what speed? How far is it to the other side? Classification: Safe to walk or not!!!
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Discriminant Function
It can be arbitrary functions of x, such as: Nearest Neighbor Decision Tree Linear Functions Nonlinear Functions
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Background – Classification Problem
Applications: Personal Identification Credit Rating Medical Diagnosis Text Categorization Denial of Service Detection Character recognition Biometrics Image classification
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Classification Formulation
Given an input space a set of classes ={ } the Classification Problem is to define a mapping f: g where each x in is assigned to one class This mapping function is called a Decision Function
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Decision Function The basic problem in classification problem is to find c decision functions with the property that, if a pattern x belongs to class i, then is some similarity measure between x and class i, such as distance or probability concept
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Decision Function Example d1=d3 Class 1 d2,d3<d1 Class 3
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Single Classifier Most popular single classifiers:
Minimum Distance Classifier Bayes Classifier K-Nearest Neighbor Decision Tree Neural Network Support Vector Machine
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SVM Support Vector Machines (SVM) The one with largest margin!!
(Separable case) Which is the best separation hyperplane? The one with largest margin!!
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Linearly Separable Classes
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Support Vector Machine
Basically a 2-class classifier developed by Vapnik and Chervonenkis (1992) Which line is optimal?
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large margin provides better generalization ability
Support Vector Machines (SVM) large margin provides better generalization ability Maximizing Margin: Correct Separation:
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Why named “Support Vector Machine”?
Support Vectors
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Support Vector Machine
Training vectors : xi , i=1….n Consider a simple case with two classes : Define a vector y yi = 1 if xi in class 1 = -1 if xi in class 2 A hyperplane which separates all data r ρ Separating plane Margin Class 1 Class 2 Support Vector (Class 1) Support Vector (Class 2)
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2.8 SVM
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Linear Separable SVM Label the training data
Suppose we have some hyperplanes which separates the “+” from “-” examples (a separating hyperplane) x which lie on the hyperplane, satisfy w is noraml to hyperplane, |b|/||w|| is the perpendicular distance from hyperplane to origin
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Linear Separable SVM Margin = distance between H1 and H2 = 2/||w||
Define two support hyperplane as H1:wTx = b +δ and H2:wTx = b –δ To solve over-parameterized problem, set δ=1 Define the distance as Margin = distance between H1 and H2 = 2/||w||
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The Primal problem of SVM
Goal: Find a separating hyperplane with largest margin. A SVM is to find w and b that satisfy (1) minimize ||w||/2 = wTw/2 (2) yi(xi·w+b)-1 ≥ 0 Switch the above problem to a Lagrangian formulation for two reason (1) easier to handle by transforming into quadratic eq. (2) training data only appear in form of dot products between vectors => can be generalized to nonlinear case
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Langrange Muliplier Method
a method to find the extremum of a multivariate function f(x1,x2,…xn) subject to the constraint g(x1,x2,…xn) = 0 For an extremum of f to exist on g, the gradient of f must line up with the gradient of g . for all k = 1, ...,n , where the constant λis called the Lagrange multiplier The Lagrangian transformation of the problem is
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Langrange Muliplier Method
To have , we need to find the gradient of L with respect to w and b. (1) (2) Substitute them into Lagrangian form, we have a dual problem Inner product form => Can be generalize to nonlinear case by applying kernel
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KKT Conditions w is determined by training procedure.
Since the problems for SVM is convex, the KKT conditions are necessary and sufficient for w, b and α to be a solution. w is determined by training procedure. b is easily found by using KKT complementary conditions, by choosing any i for which αi≠ 0 Complementary slackness
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2.8 SVM What about non-linear boundary?
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Non-Linear Separable SVM : Kernal
To extend to non-linear case, we need to the data to some other Euclidean space.
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Kernal Φ is a mapping function.
Since the training algorithm only depend on data thru dot products. We can use a “kernal function” K such that One commonly used example is radial based function (RBF) A RBF is a real-valued function whose value depends only on the distance from the origin, so that Φ(x)= Φ(||x||) ; or alternatively on the distance from some other point c, called a center, so that Φ(x,c)= Φ(||x-c||).
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Non-separable SVM Real world application usually have no OSH. We need to add an error term ζ. => To give penalty to error term, define New Lagrangian form is
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Non-separable SVM New KKT Conditions
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