Ridge function

In mathematics, a ridge function is any function $$f:\R^d\rightarrow\R$$ that can be written as the composition of a univariate function with an affine transformation, that is: $$f(\boldsymbol{x}) = g(\boldsymbol{x}\cdot \boldsymbol{a})$$ for some $$g:\R\rightarrow\R$$ and $$\boldsymbol{a}\in\R^d$$. 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 dimensionality, 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


 * $$f\left(\boldsymbol{x} + \sum_{k=1}^{d-1}c_k\boldsymbol{a}_k\right)=g\left(\boldsymbol{x}\cdot\boldsymbol{a} + \sum_{k=1}^{d-1} c_k\boldsymbol{a}_k\cdot\boldsymbol{a}\right)=g\left(\boldsymbol{x}\cdot\boldsymbol{a} + \sum_{k=1}^{d-1} c_k0\right) = g(\boldsymbol{x} \cdot \boldsymbol{a})=f(\boldsymbol{x})$$

for all $$c_i\in\R,1\le i<d$$. In other words, any shift of $$\boldsymbol{x}$$ in a direction perpendicular to $$\boldsymbol{a}$$ 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.