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 be some instance space and let = {x1, . . .,xm} be a set of points in X. Consider a feature mapping ψ → V, where is some Hilbert space (possibly of infinite dimension). Let × be a kernel function, that is, k(x,x_) = _ψ(x)(x_)_. Kernel PCA is the process of mapping the elements in 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 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 and B, verify that their product can be computed. Similarly, if an eigenvalue decomposition of some matrix is required, verify that this decomposition can be computed.

find the cost of your paper

Suggest a modification of the binary search algorithm that emulates this strategy for a list of names.

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Explain why insertion sort works well on partially sorted lists.

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Draw a class diagram that shows the relationships among the classes in this new version of the system

Jack decides to rework the banking system, which already includes the classes BankView, Bank, SavingsAccount, and RestrictedSavingsAccount. He wants to add another class for checking accounts. He sees that savings….