Supplemental References

Allan Pinkus

doi:10.1017/CBO9781316408124.016

Supplemental References

Published online by Cambridge University Press: 05 August 2015

Allan Pinkus

Show author details

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

Book contents

Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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

DOI: https://doi.org/10.1017/CBO9781316408124.016 [Opens in a new window]

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

Book contents

Supplemental References

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive