zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory. (English) Zbl 1204.68157
Summary: We give several properties of the reproducing kernel Hilbert space induced by the Gaussian kernel, along with their implications for recent results in the complexity of the regularized least square algorithm in learning theory.
MSC:
68T05Learning and adaptive systems
68P30Coding and information theory (theory of data)
References:
[1]Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950) · doi:10.1090/S0002-9947-1950-0051437-7
[2]Boucheron, S., Bousquet, O., Lugosi, G.: Theory of classification: a survey of recent advances. ESAIM: Prob. Stat. 9, 323–375 (2005) · Zbl 1136.62355 · doi:10.1051/ps:2005018
[3]Carmeli, C., De Vito, E., Toigo, A.: Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. Anal. Appl. 4, 377–408 (2006) · Zbl 1116.46019 · doi:10.1142/S0219530506000838
[4]Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Am. Math. Soc. 39(1), 1–49 (2002) · Zbl 0983.68162 · doi:10.1090/S0273-0979-01-00923-5
[5]De Vito, E., Caponnetto, A., Rosasco, L.: Model selection for regularized least-squares algorithm in learning theory. Found. Comput. Math. 5(1), 59–85 (2005) · Zbl 1083.68106 · doi:10.1007/s10208-004-0134-1
[6]Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, Products, 6th edn. Academic Press, San Diego (2000)
[7]Mercer, J.: Functions of positive and negative type, and their connection with the theory of integral equations. Philos. Trans. R. Soc. Lond., Ser. A 209, 415–446 (1909) · doi:10.1098/rsta.1909.0016
[8]Minh, H.Q.: The regularized least square algorithm and the problem of learning halfspaces. Submitted preprint (2007)
[9]Müller, C.: Analysis of Spherical Symmetries in Euclidean Spaces. Applied Mathematical Sciences, vol. 129. Springer, New York (1997)
[10]Niyogi, P., Girosi, F.: Generalization bounds for function approximation from scattered noisy data. Adv. Comput. Math. 10, 51–80 (1999) · Zbl 1053.65506 · doi:10.1023/A:1018966213079
[11]Poggio, T., Smale, S.: The mathematics of learning: dealing with data. Not. Am. Math. Soc. 50(5), 537–544 (2003)
[12]Schölkopf, B., Smola, A.J.: Learning with Kernels. MIT Press, Cambridge (2002)
[13]Smale, S., Zhou, D.X.: Learning theory estimates via integral operators and their approximations. Constr. Approx. 26(2), 153–172 (2007) · Zbl 1127.68088 · doi:10.1007/s00365-006-0659-y
[14]Steinwart, I.: On the influence of the kernel on the consistency of support vector machines. J. Mach. Learn. Res. 2, 67–93 (2001) · Zbl 1009.68143 · doi:10.1162/153244302760185252
[15]Steinwart, I., Hush, D., Scovel, C.: An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels. IEEE Trans. Inf. Theory 52, 4635–4643 (2006) · Zbl 05455266 · doi:10.1109/TIT.2006.881713
[16]Sun, H.W.: Mercer theorem for RKHS on noncompact sets. J. Complex. 21, 337–349 (2005) · Zbl 1094.46021 · doi:10.1016/j.jco.2004.09.002
[17]Sun, H.W., Zhou, D.X.: Reproducing kernel Hilbert spaces associated with analytic translation-invariant Mercer kernels. J. Fourier Anal. Appl. 14, 89–101 (2008) · Zbl 1153.46017 · doi:10.1007/s00041-007-9003-z
[18]Temlyakov, V.N.: Approximation in learning theory. Constr. Approx. 27, 33–74 (2008) · Zbl 05264756 · doi:10.1007/s00365-006-0655-2
[19]Tsybakov, A.B.: Optimal aggregation of classifiers in statistical learning. Ann. Stat. 32(1), 135–166 (2004) · Zbl 1105.62353 · doi:10.1214/aos/1079120131
[20]Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)
[21]Wahba, G.: Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1990)
[22]Yao, Y.: Early stopping in gradient descent learning. Constr. Approx. 26(2), 289–315 (2007) · Zbl 1125.62035 · doi:10.1007/s00365-006-0663-2
[23]Ying, Y., Zhou, D.X.: Learnability of Gaussians with flexible variances. J. Mach. Learn. Res. 8, 249–276 (2007)