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Some Problems in the Theory of Ridge Functions

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Abstract

Let d ≥ 2 and \(E\subset\mathbb{R}^d\) be a set. A ridge function on E is a function of the form φ(a · x), where \(x=(x_1,...,x_d)\in{E},\;a=(a_1,...,a_d)\in\mathbb{R}^d\;\backslash\left\{0\right\},\;a \cdot x = \sum\nolimits_{j = 1}^d {{a_j}{x_j}}\), and φ is a real-valued function. Ridge functions play an important role both in approximation theory and mathematical physics and in the solution of applied problems. The present paper is of survey character. It addresses the problems of representation and approximation of multidimensional functions by finite sums of ridge functions. Analogs and generalizations of ridge functions are also considered.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. V. Konyagin

  2. Laboratory “Multidimensional Approximation and Applications,”, Moscow State University, Moscow, 119991, Russia

    A. A. Kuleshov

  3. Technion – Israel Institute of Technology, Haifa, 32000, Israel

    V. E. Maiorov

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  1. S. V. Konyagin
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  2. A. A. Kuleshov
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  3. V. E. Maiorov
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Original Russian Text © S.V. Konyagin, A.A. Kuleshov, V.E. Maiorov, 2018, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 301, pp. 155–181.

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Konyagin, S.V., Kuleshov, A.A. & Maiorov, V.E. Some Problems in the Theory of Ridge Functions. Proc. Steklov Inst. Math. 301, 144–169 (2018). https://doi.org/10.1134/S0081543818040120

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