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An Introduction to Support Vector Machine (SVM) Presenter : Ahey Date : 2007/07/20 The slides are based on lecture notes of Prof. 林智仁 and Daniel Yeung
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Outline Background Linear Separable SVM Lagrange Multiplier Method Karush-Kuhn-Tucker (KKT) Conditions Non-linear SVM: Kernel Non-Separable SVM libsvm
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Background – Classification Problem The goal of classification is to organize and categorize data into distinct classes A model is first created based on the previous data (training samples) This model is then used to classify new data (unseen samples) A sample is characterized by a set of features Classification is essentially finding the best boundary between classes
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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: 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 d 3 =d 2 d 1 =d 3 d 1 =d 2 d 1,d 3 <d 2 Class 1 Class 3 Class 2 d 2,d 3 <d 1 d 1,d 2 <d 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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Minimum Distance Classifier Simplest approach to selection of decision boundaries Each class is represented by a prototype (or mean) vector: where = the number of pattern vectors from A new unlabelled sample is assigned to a class whose prototype is closest to the sample
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Bayes Classifier Bayes rule is the same for each class, therefore Assign x to class j if for all i
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Bayes Classifier The following information must be known: The probability density functions of the patterns in each class The probability of occurrence of each class Training samples may be used to obtain estimations on these probability functions Samples assumed to follow a known distribution pattern
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K-Nearest Neighbor K-Nearest Neighbor Rule (k-NNR) Examine the labels of the k-nearest samples and classify by using a majority voting scheme 0246810 2 4 6 8 (7, 3) 1NN 3NN 5NN 7NN 9NN
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Decision Tree The decision boundaries are hyper-planes parallel to the feature-axis A sequential classification procedure may be developed by considering successive partitions of R
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Decision Trees Example
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Neural Network A Neural Network generally maps a set of inputs to a set of outputs Number of inputs/outputs vary The network itself is composed of an arbitrary number of nodes with an arbitrary topology It is an universal approximator Node Connection
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Neural Network A popular NN is the feed forward neural network E.g. Multi-layer Perceptron (MLP) Radial-Based Function (RBF) Learning algorithm: back propagation Weights of nodes are adjusted based on how well the current weights match an objective
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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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Support Vector Machine Training vectors : x i, i=1….n Consider a simple case with two classes : Define a vector y y i = 1 if x i in class 1 = -1 if x i 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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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 Define two support hyperplane as H1:w T x = b +δ and H2:w T x = b –δ To solve over-parameterized problem, set δ=1 Define the distance between OSH and two support hyperplanes 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 = w T w/2 (2) y i (x i ·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(x 1,x 2,…x n ) subject to the constraint g(x 1,x 2,…x n ) = 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 Since the problems for SVM is convex, the KKT conditions are necessary and sufficient for w, b and αto be a solution. w is determinded by training procedure. b is easily found by using KKT complementary conditions, by choosing any i for which α i ≠ 0 Complementary slackness
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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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