Ridge function

Template:Distinguish In mathematics, a ridge function is any function $f:{ℝ}^d\to ℝ$ that can be written as the composition of an univariate function $g:ℝ\to ℝ$, that is called a profile function, with an affine transformation, given by a direction vector $a\in {ℝ}^d$ with shift $b\in ℝ$.

Then, the ridge function reads $f(x)=g(x^{\mathrm{\top }}a+b)$ for $x\in {ℝ}^d$.

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 dimensionalityTemplate:Clarification needed, making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in $d-1$ directions: Let $a_1,\dots ,a_{d-1}$ be $d-1$ independent vectors that are orthogonal to $a$, such that these vectors span $d-1$ dimensions. Then

$f(𝒙+\sum _{k=1}^{d-1}{c}_k{𝒂}_k)=g(𝒙\cdot 𝒂+\sum _{k=1}^{d-1}{c}_k{𝒂}_k\cdot 𝒂)=g(𝒙\cdot 𝒂+\sum _{k=1}^{d-1}{c}_{k}0)=g(𝒙\cdot 𝒂)=f(𝒙)$

for all $c_i\in ℝ,1\le i<d$. In other words, any shift of $𝒙$ in a direction perpendicular to $𝒂$ does not change the value of $f$.

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.