# Ridge function

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Ridge_function
> Markdown URL: https://mediated.wiki/source/Ridge_function.md
> Source: https://en.wikipedia.org/wiki/Ridge_function
> Source revision: 1306592292
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Distinguish|ridge regression}}
In [mathematics](/source/mathematics), a '''ridge function''' is any function <math>f:\R^d\rightarrow\R</math> that can be written as the composition of an [univariate](/source/univariate) function <math>g:\R \rightarrow\R</math>, that is called a '''profile function''', with an [affine transformation](/source/affine_transformation), given by a '''direction vector''' <math>a \in \R^d</math> with '''shift''' <math>b \in \R </math>.

Then, the ridge function reads
<math>f(x) = g(x^{\top} a +  b )</math> for <math>x\in\R^d</math>. 

Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.<ref>{{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}}</ref>

== Relevance ==
A ridge function is not susceptible to the [curse of dimensionality](/source/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 <math>d-1</math> directions:
Let <math>a_1,\dots,a_{d-1}</math> be <math>d-1</math> independent vectors that are orthogonal to <math>a</math>, such that these vectors span <math>d-1</math> dimensions.
Then 

: <math>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})</math> 

for all <math>c_i\in\R,1\le i<d</math>.
In other words, any shift of <math>\boldsymbol{x}</math> in a direction perpendicular to <math>\boldsymbol{a}</math> does not change the value of <math>f</math>.

Ridge functions play an essential role in amongst others [projection pursuit](/source/projection_pursuit), [generalized linear models](/source/generalized_linear_model), and as [activation functions](/source/activation_function) in [neural networks](/source/neural_network). For a survey on ridge functions, see.<ref>{{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 }}</ref> For books on ridge functions, see.<ref name= pinkus>{{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 }}</ref><ref name= ismailov>{{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}}</ref>

== References ==
<!-- Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. -->
{{reflist}}

Category:Functions and mappings

---
Adapted from the Wikipedia article [Ridge function](https://en.wikipedia.org/wiki/Ridge_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Ridge_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.