Support Vector Machine (continued)
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1 Support Vector Machine continued) Overlapping class distribution: In practice the class-conditional distributions may overlap, so that the training data points are no longer linearly separable. We need a way to modify support vector machine to allow some of the training data points to be misclassified. To do this, we introduce slack variables ξ n 0 for each data points. These are defined as ξ n = 0 for data points that are on or inside the correct margin boundary. ξ n = t n yx n ) for all the other data points. A data point that is on the decision boundary ξ n = 1, and data points with ξ n > 1 will be misclassified. The classification constraints are replaced with t n w T ϕx) + b ) 1 ξ n, n = 1,..., N. 1) in which the slack variables are constrained to satisfy ξ n 0. This is called soft margin. 1
2 Data points for which 0 < ξ n 1 lie inside the margin, but on the correct side of the decision boundary. Data points for which ξ n > 1 lie on the wrong side of the decision boundary. This framework allows for overlapping class distribution. Our goal is to maximize the margin while softly penalizing points that lie on the wrong side of the margin boundary. We therefore minimize C ξ n w 2 2) 2
3 where C > 0 control the trade-off between the slack variable penalty and the margin. We minimize 2) subject to the constraints 1) together with ξ n 0. The corresponding Lagrangian is given by Lw, b, ξ, a, µ) = 1 2 w 2 + C ξ n a n {t n w T ϕx) + b ) 1 + ξ n } µ n ξ n 3) where {a n 0 and µ n 0} are Lagrange multipliers. The corresponding KKT conditions are a n 0 t n w T ϕx) + b ) 1 + ξ n 0 a n {t n w T ϕx) + b ) 1 + ξ n } = 0 µ n 0 ξ n 0 µ n ξ n = 0 3
4 We now optimize out w, b and {ξ n } to give L w = 0 w = N a n t n ϕx n ) L b = 0 N a n t n = 0 L ξ n = 0 a n = C µ n 4) Using these result to eliminate w, b and {ξ n } from the Lagrangian, we obtain the dual Lagrangian in the form La) = a n 1 a n a m t n t m kx n, x m ) 2 m=1 5) which is identical to the separable case, except the constraints are different. These are 0 a n C 6) a n t n = 0 7) for n = 1,...N, where 6) are known are box constraints. This again represents a quadratic programming problem. 4
5 As before, a subset of the data points may have a n = 0, the remaining data points constitute the support vectors. These have a n = 0, and must satisfy t n yx n ) = 1 ξ n 8) If a n < C, then µ n = 0, requiring ξ n = 0, and hence these points are on the margin. Points with a n = C can lie inside the margin, and can either be correctly classified if ξ n 1, or misclassified if ξ n > 1. The predictions for new data points are made in the same way: yx) = a n t n kx, x n ) + b 9) To determine the parameter b, we note for those support vectors with 0 < a n < C, t n yx n ) = 1, hence we have b = 1 N M n M t n m S a m t m kx n, x m ) ) 5 10)
6 where M denotes the set of indices of data points having 0 < a n < C. Direct solution of this QP problem is often infeasible so more practical approaches are needed. One of the most popular approaches is called the sequential minimal optimization SMO). kernel trick: The support vector machine can learn a nonlinear decision boundary that is linear in the feature space ϕx), which can be high or infinite dimension. However, we do not need explicitly know this ϕx), but only need to define the kernels kx i, x j ). Popular kernels functions kx, y) are Polynomial kernel x T y + c) d, c 0 RBF kernel exp x y 2 /2σ 2 ) ) Sigmoid kernel tanhcx T y + d) 6
7 Summary of SVM classifier: In patter recognition SVM minimises approximately) the number of training points within the margin of separation between examples of two classes transformation of nonlinear separation into linear separation problem in the feature space kernel trick convex quadratic) optimisation 7
8 Lagrangian multipliers method not hindered by the curse of dimensionality duality sparsness - support vectors number of selected support vectors can still be large) principled handling of model complexity slower than other nets 8
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