Ridge function

In mathematics, a ridge function is any function ${\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ that can be written as the composition of a univariate function with an affine transformation, that is: ${\displaystyle f({\boldsymbol {x}})=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})}$ for some ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ and ${\displaystyle {\boldsymbol {a}}\in \mathbb {R} ^{d}}$. Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.^[1]

Relevance

A ridge function is not susceptible to the curse of dimensionality^{[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 ${\displaystyle d-1}$ directions: Let ${\displaystyle a_{1},\dots ,a_{d-1}}$ be ${\displaystyle d-1}$ independent vectors that are orthogonal to ${\displaystyle a}$, such that these vectors span ${\displaystyle d-1}$ dimensions. Then

${\displaystyle f\left({\boldsymbol {x}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\cdot {\boldsymbol {a}}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}0\right)=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})=f({\boldsymbol {x}})}$

for all ${\displaystyle c_{i}\in \mathbb {R} ,1\leq i<d}$. In other words, any shift of ${\displaystyle {\boldsymbol {x}}}$ in a direction perpendicular to ${\displaystyle {\boldsymbol {a}}}$ does not change the value of ${\displaystyle f}$.

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.^[2] For books on ridge functions, see.^[3][4]

References