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.
