Ridge function

In mathematics, a ridge function is any function that can be written as the composition of a univariate function with an affine transformation, that is: for some and . 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, 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.[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.
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