ridge function

In mathematics, a ridge function is any function $f:\R^d\rightarrow\R$ that can be written as the composition of an univariate function $g:\R \rightarrow\R$ , that is called a profile function, with an affine transformation, given by a direction vector $a \in \R^d$ with shift $b \in \R$ .

Then, the ridge function reads

$f(x) = g(x^{\top} a + b )$ for $x\in\R^d$ .

Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.{{cite journal |last1=Logan |first1=B.F. |last2=Shepp |first2=L.A. |title=Optimal reconstruction of a function from its projections |journal=Duke Mathematical Journal |date=1975 |volume=42 |issue=4 |pages=645–659 |doi=10.1215/S0012-7094-75-04256-8}}

Relevance

A ridge function is not susceptible to the curse of dimensionality{{clarification needed|reason=In what sense?|date=May 2022}}, 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.{{cite journal |last1=Konyagin |first1=S.V. |last2=Kuleshov |first2=A.A. |last3=Maiorov |first3=V.E. |title=Some Problems in the Theory of Ridge Functions |journal=Proc. Steklov Inst. Math. |date=2018 |volume=301 |pages=144–169 |doi=10.1134/S0081543818040120|s2cid=126211876 }} For books on ridge functions, see.{{Cite book|last1=Pinkus|first1=Allan|date=August 2015|title= Ridge functions|url= https://www.cambridge.org/core/books/ridge-functions/25F7FDD1F852BE0F5D29171078BA5647|location= Cambridge |publisher= Cambridge Tracts in Mathematics 205. Cambridge University Press. 215 pp. |isbn= 9781316408124 }}{{Cite book|last1=Ismailov|first1=Vugar|date=December 2021|title= Ridge functions and applications in neural networks|url= https://www.ams.org/books/surv/263/ |location= Providence, RI |publisher= Mathematical Surveys and Monographs 263. American Mathematical Society. 186 pp. |isbn=978-1-4704-6765-4}}

References

Category:Functions and mappings