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 In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...

, that is:

f(\boldsymbol) = g(\boldsymbol\cdot \boldsymbol)

for some

g:\R\rightarrow\R

and

\boldsymbol\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

independent vectors that are orthogonal to

a

, such that these vectors span

d-1

dimensions. Then :

f\left(\boldsymbol + \sum_^c_k\boldsymbol_k\right)=g\left(\boldsymbol\cdot\boldsymbol + \sum_^ c_k\boldsymbol_k\cdot\boldsymbol\right)=g\left(\boldsymbol\cdot\boldsymbol + \sum_^ c_k0\right) = g(\boldsymbol \cdot \boldsymbol)=f(\boldsymbol)

for all

c_i\in\R,1\le i .
In other words, any shift of \boldsymbol in a direction perpendicular to \boldsymbol does not change the value of f .

Ridge functions play an essential role in amongst others

projection pursuit Projection pursuit (PP) is a type of statistical technique which involves finding the most "interesting" possible projections in multidimensional data. Often, projections which deviate more from a normal distribution are considered to be more inter ...

generalized linear models In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and b ...

, and as activation functions in

neural networks A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

. For a survey on ridge functions, see. For books on ridge functions, see.

References

{{reflist Functions and mappings