lecture18 - Linear Separation of Data Choosing a Separating...

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e62: lecture 18 17/11/10 Linear Separation of Data Data Positive samples Negative samples Linear separation Linear program 1 u 1 , . . . , u M ∈± N v 1 , . . . , v K N min 0 s . t .x T u m - α 1 m =1 ,...,M - x T v k + α 1 k ,...,K x T u m - α > 0 m - x T v k + α > 0 k e62: lecture 18 17/11/10 Choosing a Separating Hyperplane 2 Idea: maximize the margin margin = distance from hyperplane to closest data point e62: lecture 18 17/11/10 Margin 3 x γ x u v x T z - α =0 - γ x θ = x T ( z + γ x ) - α θ = γ x T x θ = γ ± x ± 2 γ ± x ± = θ / ± x ± γ ± x ± =( x T u - α ) / ± x ± How to express the distance to a data point? Let θ = x T u - α θ = - x T ( z - γ x ) - α θ = γ x T x θ = γ ± x ± 2 γ ± x ± = θ / ± x ± γ ± x ± - x T v + α ) / ± x ± Let θ = - x T v + α distance = γ ± x ± margin = min data points (distance to data point) e62: lecture 18 17/11/10 Margin 4 margin( x, α ) = max θ / ± x ± s . t T u m - α θ m - x T v k + α θ k margin( x, α ) = min ± min m x T u m - α ± x ± , min k - x T v k + α ± x ± ² (maximizing over θ ) margin = min data points (distance to data point)
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e62: lecture 18 17/11/10 Maximizing the Margin 5 ( maximizing over x ∈± N , α , θ ) max θ / ± x ± s . t .x T u m - α θ m =1 ,...,M - x T v k - α θ k ,...,K max margin( x, α ) s . t T u m - α > 0 m - x T v k - α > 0 k (if θ > 0 then ( x, α ) identiFes a separating hyperplane) e62: lecture 18 17/11/10 Quadratic Program 6 Let ˜ x = x/
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lecture18 - Linear Separation of Data Choosing a Separating...

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