4 - Identifying Functions and Directions

Summary

In this chapter we consider the following inverse problem. Assume that we are given a function F that we know is of the form

for some choice of positive integer r, unknown functions f_i, and either known or unknown directions aⁱ. The question we ask is how to determine these unknowns parameters based on our knowledge of F. In the first section we assume that we know the directions aⁱ, while in the second section we assume they are unknown. In Section 4.3 we pose these same questions for generalized ridge functions. In the case of unknown Aⁱ we are able only to analyze the case r = 1. Thematerial of Sections 4.1 and 4.2 may be found in Buhmann and Pinkus [1999].

Known Directions

Assume that we know an F of the form (4.1) with given directions aⁱ. How can we theoretically identify the functions fⁱ? (We will, of course, assume that the directions aⁱ are pairwise linearly independent.) As we have seen, from the previous chapters, we have a degree of non-unicity. However, assuming that F is smooth and f_i ∊ B for all i, then from Theorem 3.1 the f_i are determined, at the very least, up to polynomials of degree at most r − 2.

Let us now detail how we might determine the f_i. When r = 1 we need make no assumptions as

F(x) = f₁(a¹ · x).

Choosing c ∊ Rⁿ such that a1 · c = 1, we have

F(tc) = f₁(t),

which gives us f₁. Similarly, for r = 2 we can find a c ∊ Rⁿ satisfying a¹ · c = 1 and a² · c = 0, and thus

F(tc) = f₁(t) + f₂(0),

which determines f₁ up to a constant. In this same manner we determine f₂ up to a constant.