Uniqueness of Linear Combinations of Ridge Functions

Jinling Long,Dong Nan,Wei Wu,Junfang Wang

doi:10.1109/icnc.2007.790

Abstract

Ridge functions are multivariate functions of the form g(a ldr x), where g is a univariate function, and a ldr x is the inner product of a isin R d \{0} and x isin R d . We are concerned with the uniqueness of representation of a given function as some sum of ridge functions. We prove that if f(x) = Sigma i=1 m g i (a i ldr x) = 0 for some a i = (a 1 i , hellip , a d i ) isin R d \{0}, and if g i isin L loc p (R) (or g i isin D' (R) and g i (a i ldr x) isin D' (R d )), then, each g i is a polynomial of degree at most m - 2. We also prove a theorem on the smoothness of linear combinations of ridge functions. These results improve the existing results.

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