In this problem, we consider constructing new kernels by combining existing kernels. Recall that for some...

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In this problem, we consider constructing new kernels by combining existing kernels. Recall that for some function k(x, z) to be a kernel, we need to be able to write it as a dot product of vectors in some high-dimensional feature space defined by o: k(r, z) = 6(r)o(z) Mercer's theorem gives a necessary and sufficient condition for a function k to be a kernel function: its corresponding kernel matrix K has to be symmetric and positive semidefinite. Suppose that k1 (2, z) and k2(r, z) are two valid kernels. For each of the cases below, state whether k is also a valid kernel. If it is, prove it. If it is not, give a counterexample. You can use either Mercer's theorem, or the definition of a kernel as needed to prove it (If you use any properties on page 10 of Lecture 8, we need to prove them first). (c) [10 points] If k(r, z) = e is a valid kernel, prove that the Gaussian kernel k(x, z) = e is 202 also a valid kernel. In this problem, we consider constructing new kernels by combining existing kernels. Recall that for some function k(x, z) to be a kernel, we need to be able to write it as a dot product of vectors in some high-dimensional feature space defined by o: k(r, z) = 6(r)o(z) Mercer's theorem gives a necessary and sufficient condition for a function k to be a kernel function: its corresponding kernel matrix K has to be symmetric and positive semidefinite. Suppose that k1 (2, z) and k2(r, z) are two valid kernels. For each of the cases below, state whether k is also a valid kernel. If it is, prove it. If it is not, give a counterexample. You can use either Mercer's theorem, or the definition of a kernel as needed to prove it (If you use any properties on page 10 of Lecture 8, we need to prove them first). (c) [10 points] If k(r, z) = e is a valid kernel, prove that the Gaussian kernel k(x, z) = e is 202 also a valid kernel.

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