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12 - chap12-slides - Rseaux SVM Marc Parizeau...

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GIF-21410/64326 Réseaux de neurones Réseaux SVM Marc Parizeau
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GIF-21410/64326 Réseaux de neurones Approche Classifieur linéaire Maximiser la marge entre les vecteurs de support Appliquer la méthode des multiplicateurs de Lagrange Usage d’un noyau non-linéaire 2
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GIF-21410/64326 Réseaux de neurones Problème simple à 2 classes 3 f : R -→ { - 1 , +1 } ( p 1 , d 1 ) , ( p 2 , d 2 ) , . . . , ( p Q , d Q ) Soit le risque R ( f ) = l ( f ( p , d ) d P ( p , d ) o`u l ( f ( p , d ) est une fonction de perte et P ( p , d ) la probabilit´ e d’observer ( p , d )
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GIF-21410/64326 Réseaux de neurones 4 Risque empirique R e = 1 Q Q i =1 l ( f ( p i ) , d i ) Q → ∞ = R e ( f ) = R ( f ) En pratique, on ne connaît pas les lois de densité
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GIF-21410/64326 Réseaux de neurones Perceptron simple 5 a Entrée Couche de S neurones a = hardlims( Wp ! b ) W b + p n -1 S x 1 R x 1 S x R S x 1 S x 1 R S
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GIF-21410/64326 Réseaux de neurones Marges et vecteurs de support marge vecteurs de support frontière de décision 6
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GIF-21410/64326 Réseaux de neurones 7 Pour minimiser le risque, il faut maximiser la marge ! On suppose que les stimuli sont linéairement séparables Soit w et b tels que | w T p q - b | 1 , q f ( p ) = w T p - b Soit p i et p j tels que w T p i - b = +1 et w T p j - b = - 1
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GIF-21410/64326 Réseaux de neurones Maximiser la marge w T p q - b 1 , q 8 w T p j = - 1 + b = δ j = - 1 + b || w || w T p i
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