Ridge function

In mathematics, a ridge function is any function [math]\displaystyle{ f:\R^d\rightarrow\R }[/math] that can be written as the composition of a univariate function with an affine transformation, that is: [math]\displaystyle{ f(\boldsymbol{x}) = g(\boldsymbol{x}\cdot \boldsymbol{a}) }[/math] for some [math]\displaystyle{ g:\R\rightarrow\R }[/math] and [math]\displaystyle{ \boldsymbol{a}\in\R^d }[/math]. Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.

Relevance

A ridge function is not susceptible to the curse of dimensionality, making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in [math]\displaystyle{ d-1 }[/math] directions: Let [math]\displaystyle{ a_1,\dots,a_{d-1} }[/math] be [math]\displaystyle{ d-1 }[/math] independent vectors that are orthogonal to [math]\displaystyle{ a }[/math], such that these vectors span [math]\displaystyle{ d-1 }[/math] dimensions. Then

[math]\displaystyle{ f\left(\boldsymbol{x} + \sum_{k=1}^{d-1}c_k\boldsymbol{a}_k\right)=g\left(\boldsymbol{x}\cdot\boldsymbol{a} + \sum_{k=1}^{d-1} c_k\boldsymbol{a}_k\cdot\boldsymbol{a}\right)=g\left(\boldsymbol{x}\cdot\boldsymbol{a} + \sum_{k=1}^{d-1} c_k0\right) = g(\boldsymbol{x} \cdot \boldsymbol{a})=f(\boldsymbol{x}) }[/math]

for all [math]\displaystyle{ c_i\in\R,1\le i\lt d }[/math]. In other words, any shift of [math]\displaystyle{ \boldsymbol{x} }[/math] in a direction perpendicular to [math]\displaystyle{ \boldsymbol{a} }[/math] does not change the value of [math]\displaystyle{ f }[/math].

Ridge functions play an essential role in amongst others projection pursuit, generalized linear models, and as activation functions in neural networks. For a survey on ridge functions, see. For books on ridge functions, see.