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. 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. == References ==