Tik-61.261 Principles of Neural Computing
Raivio, Venna

Exercise 10
1. The thin-plate-spline function is described by

 for some        and

Justify the use of this function as a translationally and rotationally invariant Green's function. Plot the function graphically. (Haykin, Problem 5.1)

2. The set of values given in Section 5.8 for the weight vector of the RBF network in Figure 5.6. presents one possible solution for the XOR problem. Solve the same problem by setting the centers of the radial-basis functions to

 and

(Haykin, Problem 5.2)

3. Consider the cost functional

which refers to the approximating function

Show that the cost functional is minimized when

where the -by- matrix , the -by- matrix , the -by-1 vector , and the -by-1 vector are defined by Equations (5.72), (5.75), (5.73), and (5.46), respectively. (Haykin, Problem 5.5)

4. Consider more closely the properties of the singular-value decomposition (SVD) discussed very briefly in Haykin, p. 300.
1. Express the matrix in terms of its singular values and vectors.
2. Show that the pseudoinverse of can be computed from Equation (5.152):

3. Show that the left and right singular vector and are obtained as eigenvectors of the matrices and , respectively, and the squared singular values are the corresponding nonzero eigenvalues.

Jarkko Venna 2005-04-13