2 - Smoothness

Summary

In this chapter we study one of the basic properties of ridge function decomposition, namely smoothness. In the first section we ask the following question. If

is smooth, does this imply that each of the f_i is also smooth? In the second section we ask this same question with regard to generalized ridge functions, i.e., linear combinations of functions of the form f(Ax), where the A are d × n real matrices, and f : R^d → R.

Ridge Function Smoothness

Let C^k(Rⁿ), k ∊ Z₊, denote the usual class of real-valued functions with all derivatives of order up to and including k being continuous. Assume F ∊ C^k(Rⁿ) is of the form

where r is finite, i.e., F ∊ M(a¹, …, a^r), and the aⁱ are given pairwise linearly independent vectors in Rⁿ. What can we say about the smoothness of the f_i? Do the f_i necessarily inherit all the smoothness properties of the F?

When r = 1 the answer is yes, and there is essentially nothing to prove. That is, if

F(x) = f₁(a¹ · x)

is in C^k(Rⁿ) for some a¹ ≠ 0, then for c ∊ Rⁿ satisfying a1 · c = 1 and all t ∊ R we have that F(tc) = f₁(t) is in C^k(R). This same result holds when r = 2. As the a¹ and a² are linearly independent, there exists a vector c ∊ Rⁿ satisfying a¹ · c = 0 and a² · c = 1. Thus

F(tc) = f₁(a¹ · tc) + f₂(a² · tc) = f₁(0) + f₂(t).

Since F(tc) is in C_k(R), as a function of t, so is f₂. The same result holds for f₁.