Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 8
139
Views
1
CrossRef citations to date
0
Altmetric
Articles

A multiscale support vector regression method on spheres with data compression

, &
Pages 1496-1519 | Received 27 Oct 2017, Accepted 14 Jan 2018, Published online: 31 Jan 2018

References

  • McEwen D , Wiaux Y , Eyers DM . Data compression on the sphere. Astron Astrophys. 2011;531:1–13.
  • Narcowich FJ , Ward JD . Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J Math Anal. 2002;33:1393–1410.
  • Le Gia QT , Sloan I , Wendland H . Multiscale analysis in Sobolev spaces on the sphere. SIAM J Numer Anal. 2010;48:2065–2090.
  • Schaback R . On the efficiency of interpolation by radial basis functions. In: Le Méhauté A , Rabut C , Schumaker LL , editors. Surface fitting and multiresolution methods. Nashville: Vanderbilt University Press; 1997. p. 309–318.
  • Floater MS , Iske A . Multistep scattered data interpolation using compactly supported radial basis functions. J Comput Appl Math. 1996;73:65–78.
  • Schaback R . Creating surfaces from scattered data using radial basis functions. In: Daehlen M , Lyche T , Schumaker LL , editors. Mathematical methods for curves and surfaces. Nashville: Vanderbilt University Press; 1995. p. 477–496.
  • Wendland H . Scattered data approximation. Cambridge: Cambridge University Press; 2005.
  • Schaback R . Multivariate interpolation and approximation by translates of a basis function. In: Chui CK , Schumaker LL , editors. Approximation theory VIII. Vol. 1, Approximation and interpolation. Singapore: World Scientific Publishing; 1995. p. 491–514.
  • Narcowich FJ , Schaback R , Ward JD . Multilevel interpolation and approximation. Appl Comput Harmon Anal. 1999;7:243–261.
  • Wendland H . Multiscale analysis in Sobolev space on bounded domains. Numer Math. 2010;116:493–517.
  • Le Gia QT , Sloan I , Wendland H . Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere. Appl Comput Harmon Anal. 2012;32:401–412.
  • Xu BX , Lu S , Zhong M . Multiscale support vector regression method in Sobolev spaces on bounded domains. Appl Anal. 2015;94:548–569.
  • Le Gia QT , Sloan I , Wendland H . Multiscale RBF collocation for solving PDEs on spheres. Numer Math. 2012;121:99–125.
  • Chernih A , Le Gia QT . Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs. IMA J Numer Anal. 2012;34:569–591.
  • Chernih A , Le Gia QT . Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains. ANZIAM J. 2013;54:137–152.
  • Townsend A , Wendland H . Multiscale analysis in Sobolev spaces on bounded domains with zero boundary values. IMA J Numer Anal. 2013;33:1095–1114.
  • Zhong M , Hon YC , Lu S . Multiscale analysis for ill-posed problem with support vector approach. J Sci Comput. 2015;64:317–340.
  • Zhong M , Lu S , Cheng J . Multiscale analysis for ill-posed problems with semi-discrete Tikhonov regularization. Inverse Probl. 2012;28:065019. 19 p.
  • Cao H , Pereverzyev SV , Sloan IH , et al . Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions. Appl Math Comput. 2016;273:993–1005.
  • Boser BE , Guyon IM , Vapnik VN . A training algorithm for optimal margin classifiers. In: Proceedings of 5th Annual Workshop on Computational Learning Theory; New York (NY): ACM; 1992. p. 144–152.
  • Cortes C , Vapnik V . Support-vector networks. Mach Learn. 1995;20:273–297.
  • Vapnik V . The nature of statistical learning theory. 2nd ed. New York (NY): Springer; 2000.
  • Rieger C , Zwicknagel B . Deterministic error analysis of support vector regression and related regularized kernel methods. J Mach Learn Res. 2009;10:2115–2132.
  • Le Gia QT , Wendland H . Data compression on the sphere using multiscale radial basis function approximation. Adv Comput Math. 2014;40:923–943.
  • Gronwall TH . On the degree of convergence of Laplace’s series. Trans Amer Math Soc. 1914;15:1–30.
  • Müller C . Spherical harmonics. Vol. 17, Lecture notes in mathematics. Berlin: Springer-Verlag; 1966.
  • Schoenberg IJ . Positive definite functions on spheres. Duke Math J. 1942;9:96–108.
  • Xu Y , Cheney EW . Strictly positive definite functions on spheres. Proc Am Math Soc. 1992;116:977–981.
  • Wendland H . Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math. 1995;4:389–396.
  • Wu Z . Compactly supported positive definite radial functions. Adv Comput Math. 1995;4:283–292.
  • Le Gia QT , Sloan IH , Wendland H . Multiscale analysis in Sobolev spaces on the sphere. SIAM J Numer Anal. 2010;48:2065–2090.
  • Le Gia QT , Wendland H . Data compression on the sphere using multiscale radial basis functions. Adv Comput Math. 2014;40:923–943.
  • Poggio T , Smale S . The mathematics of learning: dealing with data. Not Am Math Soc. 2003;50:537–544.
  • De Vito E , Rosasco L , Caponnetto A , et al . Learning from examples as an inverse problem. J Mach Learn Res. 2005;6:883–904.
  • Micchelli CA , Pontil M . Learning the kernel function via regulation. J Mach Learn Res. 2005;6:1099–1125.
  • Schölkopf B , Wiliamson RC , Bartlett PL . New support vector algorithms. Neural Comput. 2000;12:1207–1245.
  • Le Gia QT , Narcowich FJ , Ward JD , et al . Continuous and discrete least-square approximation by radial basis functions on spheres. J Approx Theory. 2006;143:124–133.
  • Renka RJ . Multivariate interpolation of large sets of scattered data. ACM Trans Math Softw. 1988;14:139–148.
  • Le Gia QT , McLean W . Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions. Adv Comput Math. 2012;40:353–375.
  • Saff EB , Kuijlaars ABJ . Distributing many points on a sphere. Math Intell. 1997;19:5–11.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.