On the representation by bivariate ridge functions

  • R. A. Aliev Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku and Baku State Univ., Azerbaijanи
  • A. A. Asgarova Azerbaijan Univ. Languages, Baku
  • V. E. Ismailov Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku
Keywords: Cauchy functional equation, ridge function, plane wave, representation, smoothness

Abstract

UDC 517.5

We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation.

References

J. Acz´el, Functional equations and their applications, Acad. Press, New York (1966).

R. A. Aliev, V. E. Ismailov, On a smoothness problem in ridge function representation, Adv. Appl. Math., 73 , 154 – 169 (2016), https://doi.org/10.1016/j.aam.2015.11.002 DOI: https://doi.org/10.1016/j.aam.2015.11.002

M. D. Buhmann, A. Pinkus, Identifying linear combinations of ridge functions, Adv. Appl. Math., 22 , 103 – 118 (1999), https://doi.org/10.1006/aama.1998.0623 DOI: https://doi.org/10.1006/aama.1998.0623

E. J. Cand`es, Ridgelets: estimating with ridge functions, Ann. Statist., 31 , 1561 – 1599 (2003), https://doi.org/10.1214/aos/1065705119 DOI: https://doi.org/10.1214/aos/1065705119

P. Diaconis, M. Shahshahani, On nonlinear functions of linear combinations, SIAM J. Sci. and Stat. Comput., 5 , 175 – 191 (1984), https://doi.org/10.1137/0905013 DOI: https://doi.org/10.1137/0905013

J. H. Friedman, W. Stuetzle, Projection pursuit regression, J. Amer. Statist. Assoc., 76 , 817 – 823 (1981). DOI: https://doi.org/10.1080/01621459.1981.10477729

N. J. Guliyev, V. E. Ismailov, On the approximation by single hidden layer feedforward neural networks with fixed weights, Neural Networks, 98 , 296 – 304 (2018). DOI: https://doi.org/10.1016/j.neunet.2017.12.007

V. E. Ismailov, Characterization of an extremal sum of ridge functions, J. Comput. and Appl. Math., 205 , № 1, 105 – 115 (2007), https://doi.org/10.1016/j.cam.2006.04.043 DOI: https://doi.org/10.1016/j.cam.2006.04.043

V. E. Ismailov, Approximation by neural networks with weights varying on a finite set of directions, J. Math. Anal. and Appl., 389 , № 1, 72 – 83 (2012), https://doi.org/10.1016/j.jmaa.2011.11.037 DOI: https://doi.org/10.1016/j.jmaa.2011.11.037

V. E. Ismailov, A review of some results on ridge function approximation, Azerb. J. Math., 3 , № 1, 3 – 51 (2013).

V. E. Ismailov, Approximation by ridge functions and neural networks with a bounded number of neurons, Appl. Anal., 94 , № 11, 2245 – 2260 (2015), https://doi.org/10.1080/00036811.2014.979809 DOI: https://doi.org/10.1080/00036811.2014.979809

V. E. Ismailov, Approximation by sums of ridge functions with fixed directions (in Russian), Algebra i Analiz, 28 , № 6, 20 – 69 (2016), https://doi.org/10.1090/spmj/1471 DOI: https://doi.org/10.1090/spmj/1471

V. E. Ismailov, A note on the equioscillation theorem for best ridge function approximation, Expo. Math., 35 , № 3, 343 – 349 (2017), https://doi.org/10.1016/j.exmath.2017.05.003 DOI: https://doi.org/10.1016/j.exmath.2017.05.003

V. E. Ismailov, A. Pinkus, Interpolation on lines by ridge functions, J. Approx. Theory, 175 , 91 – 113 (2013), https://doi.org/10.1016/j.jat.2013.07.010 DOI: https://doi.org/10.1016/j.jat.2013.07.010

F. John, Plane waves and spherical means applied to partial differential equations, Intersci., New York (1955).

I. Kazantsev, Tomographic reconstruction from arbitrary directions using ridge functions, Inverse Problems, 14 , 635 – 645 (1998), https://doi.org/10.1088/0266-5611/14/3/014 DOI: https://doi.org/10.1088/0266-5611/14/3/014

S. V. Konyagin, A. A. Kuleshov, On the continuity of finite sums of ridge functions (in Russian), Mat. Zametki, 98 , 308 – 309 (2015), https://doi.org/10.4213/mzm10787 DOI: https://doi.org/10.1134/S0001434615070378

S. V. Konyagin, A. A. Kuleshov, On some properties of finite sums of ridge functions defined on convex subsets of Rn (in Russian), Tr. Mat. Inst. Steklova, 293 (2016), https://doi.org/10.1134/S0371968516020138 DOI: https://doi.org/10.1134/S0081543816040131

S. V. Konyagin, A. A. Kuleshov, V E. Maiorov, Some problems in the theory of ridge functions (in Russian), Tr. Mat. Inst. Steklova, 301 (2018), https://doi.org/10.1134/S0371968518020127 DOI: https://doi.org/10.1134/S0081543818040120

A. Kro´o, On approximation by ridge functions, Constr. Approx., 13 , № 4, 447 – 460 (1997)б https://doi.org/10.1007/s003659900053 DOI: https://doi.org/10.1007/s003659900053

B. F. Logan, L. A. Shepp, Optimal reconstruction of a function from its projections, Duke Math. J., 42 , № 4, 645 – 659 (1975). DOI: https://doi.org/10.1215/S0012-7094-75-04256-8

V. E. Maiorov, On best approximation by ridge functions, J. Approx. Theory, 99 , № 1, 68 – 94 (1999), https://doi.org/10.1006/jath.1998.3304 DOI: https://doi.org/10.1006/jath.1998.3304

V. Maiorov, A. Pinkus, Lower bounds for approximation by MLP neural networks, Neurocomputing, 25 , 81 – 91(1999). DOI: https://doi.org/10.1016/S0925-2312(98)00111-8

F. Natterer, The mathematics of computerized tomography, Wiley, New York (1986). DOI: https://doi.org/10.1007/978-3-663-01409-6

P. P. Petrushev, Approximation by ridge functions and neural networks, SIAM J. Math. Anal., 30 , № 1, 155 – 189 (1998), https://doi.org/10.1137/S0036141097322959 DOI: https://doi.org/10.1137/S0036141097322959

A. Pinkus, Ridge functions, Cambridge Tracts Math., 205 , Cambridge Univ. Press, Cambridge (2015), https://doi.org/10.1017/CBO9781316408124 DOI: https://doi.org/10.1017/CBO9781316408124

A. Pinkus, Smoothness and uniqueness in ridge function representation, Indag. Math. (N.S.), 24 , № 4, 725 – 738 (2013), https://doi.org/10.1016/j.indag.2012.10.004 DOI: https://doi.org/10.1016/j.indag.2012.10.004

A. Pinkus, Approximation theory of the MLP model in neural networks, Acta Numer., 8 , 143 – 195 (1999), https://doi.org/10.1017/S0962492900002919 DOI: https://doi.org/10.1017/S0962492900002919

Published
24.05.2021
How to Cite
Aliev, R. A., A. A. Asgarova, and V. E. Ismailov. “On the Representation by Bivariate Ridge Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 5, May 2021, pp. 579 - 588, doi:10.37863/umzh.v73i5.263.
Section
Research articles