Ridge function

A ridge function is any function $f:\mathbb {R} ^{d}\rightarrow \mathbb {R}$ that can be written as the composition of a univariate function with an affiene transformation, that is: $f({\boldsymbol {x}})=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})$ for some $g:\mathbb {R} \rightarrow \mathbb {R}$ and ${\boldsymbol {a}}\in \mathbb {R} ^{d}$. 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 dimensionality, making it an instrumental tool in various estimation problems. Ridge functions play an essential role in amongst others projection pursuit, generalized linear models, and as activation functions in neural networks. A defining property of ridge functions is that they are constant in $d-1$ directions. Let $a_{1},\dots ,a_{d-1}$ be $d-1$ independent vectors that are orthogonal to $a$. Then each point Failed to parse (syntax error): {\displaystyle f\left(c_0\boldsymbol{a} + \sum_{k=1}c_k\boldsymbol{a}_k)=f\left(c_0\boldsymbol{a})}