# 1. Kernels In the lectures, we have seen that kernel functions are ...

## Transcribed Text

1. Kernels In the lectures, we have seen that kernel functions are efficient tools for computing inner products in certain types of feature spaces. In this part of the problem set we are going to investigate various properties of kernels. (a) One way to construct kernels is to build them from simpler ones. We have seen various construction rules, including the following: assuming K1 (x, 2) and K2(x,2) are kernels, then SO are (scaling) f(x)f(z)K1(xx, for any function f(x) E R (sum) K (x, 2) = = K1(x,2) - + K2(x, 2) (product) K (x,)=K1(x,)K2(x,) = Using the above rules, and the fact that K (x, z) = x.z is a kernel, show that the following is also a kernel: (b) Assume that training samples x are vectors in R2. Consider the kernel 1 What is the function such that =

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