Ridge function
In mathematics, a ridge function is any function that can be written as the composition of an univariate function, that is called a profile function, with an affine transformation, given by a direction vector with shift.
Then, the ridge function reads
for.
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 directions:Let be independent vectors that are orthogonal to, such that these vectors span dimensions.
Then
for all.
In other words, any shift of in a direction perpendicular to does not change the value of.
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.