1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## show how PCA can be used for constructing nonlinear dimensionality reduction on the basis of the kernel trick (see Chapter 16).

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*, . . .,*x*m*} 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*(x*m*)} into R*n*. 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.