Ridge function

Not to be confused with ridge regression.

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

Then, the ridge function reads ${\displaystyle f(x)=g(x^{\top }a+b)}$ for ${\displaystyle x\in \mathbb {R} ^{d}}$.

Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.[1]

Relevance

A ridge function is not susceptible to the curse of dimensionality^[clarification needed], making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in ${\displaystyle d-1}$ directions: Let ${\displaystyle a_{1},\dots ,a_{d-1}}$ be ${\displaystyle d-1}$ independent vectors that are orthogonal to ${\displaystyle a}$, such that these vectors span ${\displaystyle d-1}$ dimensions. Then

${\displaystyle f\left({\boldsymbol {x}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\cdot {\boldsymbol {a}}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}0\right)=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})=f({\boldsymbol {x}})}$

for all ${\displaystyle c_{i}\in \mathbb {R} ,1\leq i<d}$. In other words, any shift of ${\displaystyle {\boldsymbol {x}}}$ in a direction perpendicular to ${\displaystyle {\boldsymbol {a}}}$ does not change the value of ${\displaystyle f}$.

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.[2] For books on ridge functions, see.[3][4]

References

1. Logan, B.F.; Shepp, L.A. (1975). "Optimal reconstruction of a function from its projections". Duke Mathematical Journal. 42 (4): 645–659. doi:10.1215/S0012-7094-75-04256-8.
2. Konyagin, S.V.; Kuleshov, A.A.; Maiorov, V.E. (2018). "Some Problems in the Theory of Ridge Functions". Proc. Steklov Inst. Math. 301: 144–169. doi:10.1134/S0081543818040120. S2CID 126211876.
3. Pinkus, Allan (August 2015). Ridge functions. Cambridge: Cambridge Tracts in Mathematics 205. Cambridge University Press. 215 pp. ISBN 9781316408124.
4. Ismailov, Vugar (December 2021). Ridge functions and applications in neural networks. Providence, RI: Mathematical Surveys and Monographs 263. American Mathematical Society. 186 pp. ISBN 978-1-4704-6765-4.