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Title: Making Data Linearly Separable by Feature Space Mapping Consider the infinite-dimensional feature space mapping: R-R x(1[a-x<]exp-1/1-|a-x|/ OER It may be helpful to sketch the function f=1[<1-exp-1/1-for understanding the mapping and answering the questions below.) i. Show that for any n distinct points, there exists a >0 such that the mapping can linearly separate any binary labeling of the n points. ii. Show that one can efficiently compute the dot products in this feature space by giving an analytical formula for x (for arbitrary points x). iii. Given an input space X and a feature space mapping o that maps elements from X to a function that can efficiently compute the inner products in V, that is, for any X Kx,x=xx. Consider a binary classification algorithm that predicts the label of an unseen instance according to the class with the closest average. Formally, given a training set S=y1mYm for each y{1}, define Cy 11 myy=y where m={iyi=y. Assume that m+ and m- are nonzero. Then, the algorithm outputs the following decision rule: +11|-c+1|x-c-| hx= -1otherwise. a. Let w=c+-c- and let b=lc-|-|c+. Show that hx=sign(wx+b). b. Show that h can be expressed via K, without accessing individual entries of or w, thus showing that it is efficiently computable.

          Title: Making Data Linearly Separable by Feature Space Mapping

Consider the infinite-dimensional feature space mapping:
R-R x(1[a-x<]exp-1/1-|a-x|/ OER

It may be helpful to sketch the function f=1[<1-exp-1/1-for understanding the mapping and answering the questions below.)

i. Show that for any n distinct points, there exists a >0 such that the mapping can linearly separate any binary labeling of the n points.

ii. Show that one can efficiently compute the dot products in this feature space by giving an analytical formula for x (for arbitrary points x).

iii. Given an input space X and a feature space mapping o that maps elements from X to a function that can efficiently compute the inner products in V, that is, for any X Kx,x=xx. Consider a binary classification algorithm that predicts the label of an unseen instance according to the class with the closest average. Formally, given a training set S=y1mYm for each y{1}, define Cy 11 myy=y where m={iyi=y. Assume that m+ and m- are nonzero. Then, the algorithm outputs the following decision rule:
+11|-c+1|x-c-| hx= -1otherwise.

a. Let w=c+-c- and let b=lc-|-|c+. Show that hx=sign(wx+b).

b. Show that h can be expressed via K, without accessing individual entries of or w, thus showing that it is efficiently computable.

making data linearly separable by feature space mapping consider the infinite dimensional feature space mapping r r x1a xexp 11 a x oer it may be helpful to sketch the function f11 exp 11 fo 31108

Added by Anthony H.

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