lect14 - 2 || || 2 Margin w =-≤ •-≥ • = 1 b x w if...

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Lecture outline Support vector machines
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Support Vector Machines Find a linear hyperplane (decision boundary) that will separate the data
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Support Vector Machines One Possible Solution B 1
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Support Vector Machines Another possible solution B 2
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Support Vector Machines Other possible solutions B 2
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Support Vector Machines Which one is better? B1 or B2? How do you define better? B 1 B 2
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Support Vector Machines Find hyperplane maximizes the margin => B1 is better than B2 B 1 B 2 b 11 b 12 b 21 b 22 margin
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Support Vector Machines B 1 b 11 b 12 0 = + b x w 1 - = + b x w 1 + = + b x w - + - + = 1 b x w if 1 1 b x w if 1 ) ( x f 2 || || 2 Margin w =
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Support Vector Machines We want to maximize: Which is equivalent to minimizing: But subjected to the following constraints: This is a constrained optimization problem Numerical approaches to solve it (e.g., quadratic
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Unformatted text preview: 2 || || 2 Margin w = -≤ + •-≥ + • = 1 b x w if 1 1 b x w if 1 ) ( i i i x f 2 || || ) ( 2 w w L = Support Vector Machines • What if the problem is not linearly separable? Support Vector Machines • What if the problem is not linearly separable? – Introduce slack variables • Need to minimize: • Subject to: +-≤ + •-≥ + • = i i i i 1 b x w if 1-1 b x w if 1 ) ( ξ i x f + = ∑ = N i k i C w w L 1 2 2 || || ) ( Nonlinear Support Vector Machines • What if decision boundary is not linear? Nonlinear Support Vector Machines • Transform data into higher dimensional space...
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