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2 Support Vector Machine [ 3 0 pts ] Suppose we use suppor vector machines with the kernel: K ( x , x ' )

2 Support Vector Machine [30 pts]
Suppose we use suppor vector machines with the kernel:
K(x,x')={1ifx=x'0otherwise
As we discussed in class, this corresponds to mapping each x to a vector (x) in some high
dimensional space (that need not be specified) so that K(x,x')=(x)T(x')
As usual, we are given m examples (x1,y1),cdots,(xm,ym) where yiin{-1,+1}. Assume for
simplicity that all the xi's are distinct(i.e.,xixj for ij).
(1)10pts Recall that the weight vector w used in SVM's has the form
w=i?iyi(xi).
Compute the i's explicitly that would be found using SVM's with this kernel.
(2)[10 pts] Recall that the SVM algorithm outputs a classifier that, on input x, computes the sign of
w*(x). What is the value of this inner product on training example xi? What is the value of this
inner product on any example x not seen during training? Based on these answers, what kind of
generalization error do you expect will be achieved by SVM's using this kernel?
(3)[10 pts] Recall that the generalization error of SVM's can be bounded using the margin (which is
equal to 1||w||), or using the number of support vectors. What is in this case? How many support
vectors are there in this case? How are these answers consistent with your answer in part (b)?
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