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2 - Smoothness

Published online by Cambridge University Press:  05 August 2015

Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
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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 fi 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 : Rd → R.

Ridge Function Smoothness

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

where r is finite, i.e., FM(a1, …, ar), and the ai are given pairwise linearly independent vectors in Rn. What can we say about the smoothness of the fi? Do the fi 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 Ck(Rn) for some a10, then for c ∊ Rn satisfying a1 · c = 1 and all t ∊ R we have that F(tc) = f1(t) is in Ck(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 Ck(R), as a function of t, so is f2. The same result holds for f1.

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Chapter
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Ridge Functions , pp. 12 - 18
Publisher: Cambridge University Press
Print publication year: 2015

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  • Smoothness
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.004
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  • Smoothness
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.004
Available formats
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  • Smoothness
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.004
Available formats
×