This monograph is about Ridge Functions. A ridge function is any multivariate real-valued function

F : Rn → R

of the form

F(x1, …, xn) = f(a1x1 + … + anxn) = f(a · x),

where x = (x1, …, xn) ∊ Rn are the variables, f is a univariate real-valued function, i.e., f : R → R, and a = (a1, …, an) ∊ Rn\{0} is a fixed vector. This vector a ∊ Rn\{0} is generally called the direction. In other words, a ridge function is a multivariate function constant on the parallel hyperplanes a · x = c, c ∊ R. It is one of the simpler multivariate functions. Namely, it is a superposition of a univariate function with one of the simplest multivariate functions, the inner product.

More generally, we can and will consider, for given d, 1 ≤ d ≤ n−1, functions F of the form

F(x) = f(Ax),

where A is a fixed d × n real matrix, and f : Rd → R. We call such functions Generalized Ridge Functions. For d = 1, this reduces to a ridge function.

Motivation

We see specific ridge functions in numerous multivariate settings without considering them of interest in and of themselves. We find them, for example, as kernels in integral formulæ. They appear in the Fourier transform

and its inverse. We see them in the n-dimensional Radon transform

and its inverse. Here the integral is taken with respect to the natural hypersurface measure dσ. It is possible to generalize the Radon transform still further by integrating over (n − d)-dimensional affine subspaces of Rn.