In this chapter we consider the problem of the uniqueness of the representation of a linear combination of a finite number of ridge functions. That is, assume we have two distinct representations for F of the form

where both k and l are finite. What can we say about these two representations? From linearity (3.1) is effectively equivalent to asking the following. Assume

for all x ∊ Rn, where r is finite, and the ai are pairwise linearly independent vectors in Rn. What does this imply regarding the fi?

The main result of the first section of this chapter is that, with minimal requirements, the fi satisfying (3.2) must be polynomials of degree ≤ r − 2. That is,we essentially have uniqueness of the representation of a finite linear combination of ridge functions up to polynomials of a certain degree. We extend this result, in the second section, to generalized ridge functions. Much of the material of this chapter is taken from Pinkus [2013], and generalizes a result of Buhmann and Pinkus [1999].

Ridge Function Uniqueness

We recall from Chapter 2 that B is any linear space, closed under translation, of real-valued functions f defined on R such that if there is a function g ∊ C(R) for which f − g satisfies the Cauchy Functional Equation (2.2), then f − g is necessarily a linear function.

As in Section 1.3, let denote the set of algebraic polynomials of total degree at most m in n variables. That is,

Theorem 3.1Assume (3.2) holds where r is finite, and theaiare pairwise linearly independent vectors in Rn. Assume, in addition, that fi ∊ B for i = 1, …, r. Then fi is a univariate polynomial of degree at most r − 2, i = 1, …, r.