Question

# (a) Define the notion of a feature mapping in machine learning. Define the kernel corresponding to a given feature mapping. (b) Give an advantage of using kernels over applying a feature mapping explicitly. (c) Suppose K is a kernel. i. What is the corresponding normalized kernel? Make sure to avoid any possibility of dividing by zero in your formula.[4 marks] ii. Give an advantage of using normalized kernels. (d) Describe in detail the kernel form of the 1-Nearest Neighbour algorithm. (e) You are given the following training set: object (3, 1,0) labelled as 1, object (0,2,-1) labelled as -1. Using the polynomial kernel K (x,z¹) = (1+x-²)², find the 1-Nearest Neighbour prediction for object (1,1,1), showing your calculations. Consider the learning problem as classification.[7 marks] (f) Describe in detail the algorithm of Kernel Ridge Regression (KRR). Make sure to define your notation. (g) You are given the following regression training set: object (0,0,0) labelled as 1. object (0, 1,0) labelled as -1. Using the polynomial kernel K(r, x¹) = (1+x)² and ridge parameter (also known as tuning parameter) À = 1, find the KRR prediction for object (0, 0, 1).If you need to invert a 2 x 2 matrix, you may do so using the formula \left(\begin{array}{ll}

a & b \\

c & d

\end{array}\right)^{-1}=\frac{1}{a d-b c}\left(\begin{array}{cc}

d & -b \\

-c & a

\end{array}\right) .

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