(a) For any two documents z and z (note that z and z are not vectors),...

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(a) For any two documents z and z (note that z and z are not vectors), define a function k(x,z) to be the number of unique words that occur in both z andz (i.e. the size of the intersection of the sets of words in the two documents). Is this function a kernel? Justify your answer. You can assume that the size of the vocabulary is D. Hint: k(x, z) is a kernel if there exists o(r) such that k(x,z) = o(x)o(z). (b) Suppose k(x,x') is a valid kernel. Show that the exponential kernel k(x, x') = exp(k(x, x)) corresponds to a dot product in an infinite dimensional feature space. You can use the following two facts in your solution: e (where x is a scalar) can be expressed using the Taylor serie expansion as follows: k=0 Suppose k(x, x') is a valid kernel. For a polynomial q(k(x,x')), e.g. aaki (x, x')d with ad > 0, there exists a feature space Od(x) such that q(ki(x, x')) acts as a dot product in that space, i.e. q(k(x, x))=a(x)Tod(x) (a) For any two documents z and z (note that z and z are not vectors), define a function k(x,z) to be the number of unique words that occur in both z andz (i.e. the size of the intersection of the sets of words in the two documents). Is this function a kernel? Justify your answer. You can assume that the size of the vocabulary is D. Hint: k(x, z) is a kernel if there exists o(r) such that k(x,z) = o(x)o(z). (b) Suppose k(x,x') is a valid kernel. Show that the exponential kernel k(x, x') = exp(k(x, x)) corresponds to a dot product in an infinite dimensional feature space. You can use the following two facts in your solution: e (where x is a scalar) can be expressed using the Taylor serie expansion as follows: k=0 Suppose k(x, x') is a valid kernel. For a polynomial q(k(x,x')), e.g. aaki (x, x')d with ad > 0, there exists a feature space Od(x) such that q(ki(x, x')) acts as a dot product in that space, i.e. q(k(x, x))=a(x)Tod(x)