18-660: Numerical Methods for Engineering Design and Optimization
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1 8-66: Numercal Methods for Engneerng Desgn and Optmzaton n L Department of EE arnege Mellon Unversty Pttsburgh, PA 53 Slde
2 Overve lassfcaton Support vector machne Regularzaton Slde
3 lassfcaton Predct categorcal output (.e., to or multple classes) from nput attrbutes (.e., features) Example: to-class classfcaton f ( ) = < ( lass A) ( lass B) Feature x lass A lass B Feature x Slde 3
4 lassfcaton lassfcaton vs. regresson Input attrbutes Regresson Predcton of real-valued output Input attrbutes lassfcaton Predcton of categorcal output Slde 4
5 lassfcaton Examples Identfy hand-rtten dgts from US zp codes Bshop, Pattern recognton and machne learnng, 7 Slde 5
6 lassfcaton Examples Identfy geometrcal structure from ol flo data Blue: geometrcal structure Green: geometrcal structure Red: geometrcal structure 3 Bshop, Pattern recognton and machne learnng, 7 Slde 6
7 Support Vector Machne (SVM) Support vector machne (SVM) s a popular algorthm used for many classfcaton problems Key dea: maxmze classfcaton margn (mmune to nose) o-class lnear support vector machne Feature x lass B lass A Margn f ( ) = < ( lass A) ( lass B) Determne and th maxmum margn Feature x Slde 7
8 Margn alculaton o maxmze margn, e must frst represent margn as a functon of and = lass A Plus plane f ( ) = < ( lass A) ( lass B) Mnus plane lass B Support vectors Margn Plus plane = Mnus plane = (Rght-hand sde can be normalzed to ±) Slde 8
9 Margn alculaton s perpendcular to plus/mnus planes Plus plane = Mnus plane = x A A B A B = = B = ( A B) = s perpendcular to (A B) x Slde 9
10 Margn alculaton Margn equals to the dstance beteen m and p p = m λ Margn = p m = λ Fnd λ to determne margn x p λ = m = x Slde
11 Margn alculaton p = m p m λ = = ( ) = λ = p m x p λ = m = x Slde
12 Margn alculaton λ = λ = Margn = λ = λ = Maxmzng margn mples mnmzng x p λ = m = x Slde
13 Mathematcal Formulaton Start from a set of tranng samples : y : (, y ) ( =,, N ), nput feature of -th samplng pont output label of -th samplng pont lass A y = lass B y = = lass A = lass A: lass B: y y = ( ) lass B y y ( ) = Slde 3
14 Mathematcal Formulaton Formulate a convex optmzaton problem max, S.. y ( ) ( =,,, N ) Maxmze margn All data samples are n the rght class mn, S.. y ( ) ( =,,, N ) (onvex optmzaton) onvex quadratc functon Lnear constrants Slde 4
15 A Smple SVM Example o tranng samples lass A: x =, x = and y = lass B: x =, x = and y = x f ( ) = x x < ( lass A) ( lass B) Solve, and to determne classfer lass B lass A x Slde 5
16 A Smple SVM Example o tranng samples lass A: x =, x = and y = lass B: x =, x = and y = x mn, S.. y ( ) ( =,,, N ) lass B lass A x mn, S.. ( ) ( ) Slde 6
17 Slde 7 A Smple SVM Example ( ) ( ) S.. mn, S.. mn, S.. mn,.5 = = =
18 A Smple SVM Example o tranng samples lass A: x =, x = and y = lass B: x =, x = and y = x = = =.5 lass B lass A x f ( ) =.5x.5x < ( lass A) ( lass B).5x.5x = Slde 8
19 Support Vector Machne th Nose In practce, tranng samples may contan nose or are not lnearly separable mn, S.. mn,, ξ S.. y ( ) ( =,,, N ) (No feasble soluton) ( ) ( =,,, N ) Parameter determned by cross valdaton ξ λ y ξ ξ Error of -th tranng sample Feature x lass A lass B Margn Feature x Slde 9
20 Support Vector Machne th Nose an be solved by convex programmng ost : sum of to convex functons onstrants: lnear and hence convex Lnear (convex) mn,, ξ S.. y ξ ξ λ ( ) ( =,,, N ) Quadratc (convex) ξ onvex Lnear (onvex optmzaton) Slde
21 Regularzaton Regresson vs. classfcaton Regularzaton mn α Aα B Regresson λ α mn,, ξ S.. y ξ ξ λ ( ) ( =,,, N ) ξ Support vector machne Other regularzaton forms can also be used for support vector machne Slde
22 Slde Regularzaton L -norm regularzaton s used to fnd a sparse soluton of ( ) ( ) N y,,, S.. mn,, = ξ ξ λ ξ ξ L -norm regularzaton Important for feature selecton ( ) ( ) N y,,, S.. mn,, = ξ ξ λ ξ ξ
23 Slde 3 Regularzaton Feature selecton ( ) ( ) ( ) < = B lass A lass f [ ] x x x x x Important features
24 Summary lassfcaton Support vector machne Regularzaton Slde 4
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