- 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)

- 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)

- 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)

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

- 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