Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-27T21:13:21.880Z Has data issue: false hasContentIssue false

Supplemental References

Published online by Cambridge University Press:  05 August 2015

Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Ridge Functions , pp. 202 - 204
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Babayev, M.-B. A. [2004]: On estimation of the best approximation by ridge polynomials, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 20, 3–8.Google Scholar
Babayev, M.-B. A., Novruzova, N. A. [2003]: On de la Valle-Poussin type theorem, Proc.Inst. Math. Mech. Natl. Acad. Sci. Azerb. 19, 45–48.Google Scholar
Babenko, V. F., Levchenko, D. A. [2013]: Uniformly distributed ridge approximation of some classes of harmonic functions, Ukrainian Math. J. 64, 1621–1626.Google Scholar
Candès, E. J. [2002]: New ties between computational harmonic analysis and approximation theory, in Approximation Theory, X (St. Louis, MO, 2001), 87–153, eds.C. K., Chui, L. L., Schumaker, J., Stöckler, Innov. Appl. Math., Vanderbilt University Press, Nashville, TN.
Candès, E. J. [2003]: Ridgelets: estimating with ridge functions, Ann. Statist. 31, 1561–1599.Google Scholar
Cheney, E. W. [1992]: Approximation by functions of nonclassical form, in Approximation Theory, Spline Functions and Applications (Maratea, 1991), 1–18, ed. S. P.|Singh, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 356, Kluwer Academic Publishers, Dordrecht.
Cheney, E. W., Xu, Y. [1993]: A set of research problems in approximation theory, in Topics in Polynomials of One and Several Variables and their Applications, 109–123,eds. Th. M., Rassias, H. M., Srivastava, A., Yanushauskas, World Scientific Publishing.Google Scholar
Chui, C. K., Li, X. [1992]: Approximation by ridge functions and neural networks withone hidden layer, J. Approx. Theory 70, 131–141.Google Scholar
Davison, M. E., Grunbaum, F. A. [1981]: Tomographic reconstruction with arbitrary directions, Comm. Pure and Applied Math. 34, 77–120.Google Scholar
DeVore, R. A., Oskolkov, K. I., Petrushev, P. P. [1997]: Approximation by feed-forwardneural networks, The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, in Ann. Numer. Math. 4, 261–287.Google Scholar
Donoho, D. L. [2001]: Ridge functions and orthonormal ridgelets, J. Approx. Theory 111,143–179.Google Scholar
Garkavi, A. L. [1996]: On the problem of best approximation of a function f(x, y) by sums φ(ax + by) + ψ(cx + dy) (on a question of S. B. Stechkin), Proceedings of the XX Workshop on Function Theory (Moscow, 1995) in East J. Approx. 2, 151–154.Google Scholar
Garkavi, A. L., Medvedev, V. A., Khavinson, S. Ya. [1996]: Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables, Mat. Sb. 187, 3–14; English translation in Sb. Math. 187, 623–634.Google Scholar
Gordon, Y., Maiorov, V., Meyer, M., Reisner, S. [2002]: On the best approximation byridge functions in the uniform norm, Constr. Approx. 18, 61–85.Google Scholar
Hemmat, A. A., Dehghan, M. A., Skopina, M. [2005]: Ridge wavelets on the ball, J. Approx. Theory 136, 129–139.Google Scholar
Ismailov, V. E. [2006]: On the approximation by linear combinations of ridge functions inL2 metric, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 24, 101–108.Google Scholar
Ismailov, V. E. [2007c]: Representation of multivariate functions by sums of ridge functions, J. Math. Anal. Appl. 331, 184–190.Google Scholar
Ismailov, V. E. [2008b]: On the approximation by weighted ridge functions, An. Univ. Vest Timis,. Ser. Mat.-Inform. 46, 75–83.Google Scholar
Ismailov, V. E. [2011]: Approximation capabilities of neural networks with weights from two directions, Azerb. J. Math. 1, 122–128.Google Scholar
Ismailov, V. E. [2013]: A review of some results on ridge function approximation, Azerb. J. Math. 3, 3–51.Google Scholar
Jones, L. K. [2009]: Local minimax learning of functions with best finite sample estimation error bounds: applications to ridge and lasso regression, boosting, tree learning, kernel machines, and inverse problems, IEEE Trans. Inform. Theory 55, 5700–5727.Google Scholar
Kazantsev, I. G. [1998]: Tomographic reconstruction from arbitrary directions using ridge functions, Inverse Problems 14, 635–645.Google Scholar
Kolleck, A., Vybı́ral, J. [2015]: On some aspects of approximation of ridge functions, J. Approx. Theory 194, 35–61.Google Scholar
Konovalov, V. N., Kopotun, K. A., Maiorov, V. E. [2010]: Convex polynomial and ridge approximation of Lipschitz functions in Rd, Rocky Mountain J. Math. 40, 957–976.Google Scholar
Konovalov, V. N., Leviatan, D., Maiorov, V. E. [2008]: Approximation by polynomials and ridge functions of classes of s-monotone radial functions, J. Approx. Theory 152, 20–51.Google Scholar
Konovalov, V. N., Leviatan, D., Maiorov, V. E. [2009]: Approximation of Sobolev classesby polynomials and ridge functions, J. Approx. Theory 159, 97–108.Google Scholar
Kozarev, R. [2004]: The greedy ridge algorithm in Gaussian weighted L2, East J. Approx. 10 (2004), 419–440.Google Scholar
Levesley, J., Sun, X. [1995]: Scattered Hermite interpolation by ridge functions, Numer. Funct. Anal. Optim. 16, 989–1001.Google Scholar
Li, W., Padula, S. [2005]: Approximation methods for conceptual design of complexsystems, in Approximation Theory XI: Gatlinburg 2004, 241–278, eds. C. K., Chui, M., Neamtu, L. L., Schumaker, Modern Methods Math., Nashboro Press, Brentwood.
Light, W. [1993]: Ridge functions, sigmoidal functions and neural networks, in Approximation Theory VII (Austin, TX, 1992), 163–206, eds. E. W., Cheney, C. K., Chui, L. L., Schumaker, Academic Press, Boston.
Lin, V. Ya., Pinkus, A. [1994]: Approximation of multivariate functions, in Advances in Computational Mathematics, 257–266, eds. H. P., Dikshit, C. A., Micchelli, World Scientific Publishing.
Madych, W. R., Nelson, S. A. [1985]: Radial sums of ridge functions: a characterization, Math. Methods Appl. Sci. 7, 90–100.Google Scholar
Maiorov, V. [2010b]: Geometric properties of the ridge function manifold, Adv. Comput. Math. 32, 239–253.Google Scholar
Maiorov, V., Pinkus, A. [1999]: Lower bounds for approximation by MLP neural networks, Neurocomputing 25, 81–91.Google Scholar
Marshall, D. E., O'Farrell, A. G. [1983]: Approximation by a sum of two algebras. The lightning bolt principle, J. Funct. Anal. 52, 353–368.Google Scholar
Mayer, S., Ullrich, T., Vybı́ral, J. [2014]: Entropy and sampling numbers of classes of ridge functions, arXiv:1311.2005.
Oskolkov, K. I. [1999b]: Approximation by ridge functions and the Nikol'skii-Kolmogorovproblem, Dokl. Akad. Nauk 368, 445–448; English translation in Dokl. Math. 60, 209–212.Google Scholar
Park, M. G., Sun, J. [1998]: Tests in projection pursuit regression, J. Statist. Plann. Inference 75, 65–90.Google Scholar
Pelletier, B. [2004]: Approximation by ridge function fields over compact sets, J. Approx. Theory 129, 230–239.Google Scholar
Pinkus, A. [1995]: Some density problems in multivariate approximation, in Approximation Theory: Proceedings of the International Dortmund Meeting IDOMAT 95, 277–284, eds. M. W., Muller, M., Felten, D. H., Mache, Akademie Verlag.
Pinkus, A. [1997]: Approximating by ridge functions, in Surface Fitting and Multiresolution Methods, 279–292, eds. A., Le Mehaute, C., Rabut, L. L., Schumaker, Vanderbilt University Press, Nashville.
Reid, L., Sun, X. [1993]: Distance matrices and ridge function interpolation, Canad. J. Math. 45, 1313–1323.Google Scholar
Sanguineti, M. [2008]: Universal approximation by ridge computational models and neural networks: a survey, Open Appl. Math. J. 2, 31–58.Google Scholar
Sproston, J. P., Strauss, D. [1992]: Sums of subalgebras of C(X), J. London Math. Soc. 45, 265–278.Google Scholar
Sternfeld, Y. [1978]: Uniformly separating families of functions, Israel J. Math. 29, 61–91.Google Scholar
Sun, X., Cheney, E. W. [1992]: The fundamentality of sets of ridge functions, Aequationes Math. 44, 226–235.Google Scholar
Wang, Z., Qin, X., Wei G., Su, L., Wang, L. H., Fang, B. Y. [2010]: Meshless method withridge basis functions, Applied Math. Comp. 217, 1870–1886.Google Scholar
Wu, W., Feng, G., Li, X. [2002]: Training multilayer perceptrons via minimization of sum of ridge functions, Adv. Comput. Math. 17, 331–347.Google Scholar
Xu, Y., Light, W. A., Cheney, E. W. [1993]: Constructive methods of approximation by ridge functions and radial functions, Numer. Algorithms 4, 205–223.Google Scholar
Zhang, L. W. [2005]: Error estimates for interpolation with ridge basis functions, (Chinese) J. Fudan Univ. Nat. Sci. 44, 301–306.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Supplemental References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.016
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Supplemental References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.016
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Supplemental References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Ridge Functions
  • Online publication: 05 August 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316408124.016
Available formats
×