3 - Uniqueness

Summary

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 f_i?

The main result of the first section of this chapter is that, with minimal requirements, the f_i 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 f_i ∊ B for i = 1, …, r. Then f_i is a univariate polynomial of degree at most r − 2, i = 1, …, r.