1. Kernel PCA: In this exercise we show how PCA can be used for constructing nonlinear dimensionality reduction on the basis of the kernel trick (see Chapter 16). Let X be some instance space and let S = {x1, . . .,xm} be a set of points in X. Consider a feature mapping ψ : X → V, where V is some Hilbert space (possibly of infinite dimension). Let K : X × X be a kernel function, that is, k(x,x_) = _ψ(x),ψ(x_)_. Kernel PCA is the process of mapping the elements in S into V using ψ, and then applying PCA over {ψ(x1), . . .,ψ(xm)} into Rn. The output of this process is the set of reduced elements. Show how this process can be done in polynomial time in terms of m and n, assuming that each evaluation of K(·, ·) can be calculated in a constant time. In particular, if your implementation requires multiplication of two matrices A and B, verify that their product can be computed. Similarly, if an eigenvalue decomposition of some matrix C is required, verify that this decomposition can be computed.