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)
Application of integral operator for regularized least-square regression. (English) Zbl 1165.45310
Summary: We study the consistency of the regularized least-square regression in a general reproducing kernel Hilbert space. We characterize the compactness of the inclusion map from a reproducing kernel Hilbert space to the space of continuous functions and show that the capacity-based analysis by uniform covering numbers may fail in a very general setting. We prove the consistency and compute the learning rate by means of integral operator techniques. To this end, we study the properties of the integral operator. The analysis reveals that the essence of this approach is the isomorphism of the square root operator.
MSC:
45P05Integral operators
References:
[1]Alon, N.; Ben-David, S.; Cesa-Bianchi, N.; Haussler, D.: Scale-sensitive dimension, uniform convergence and learnability, J. assoc. Comput. Mach 44, 615-631 (1997) · Zbl 0891.68086 · doi:10.1145/263867.263927 · doi:http://www.acm.org/pubs/contents/journals/jacm/1997-44/
[2]Aronszajn, N.: Theory of reproducing kernels, Trans. amer. Math. soc. 68, 337-404 (1950) · Zbl 0037.20701 · doi:10.2307/1990404
[3]Bartlett, P. L.; Mendelson, S.: Rademacher and Gaussian complexities: risk bounds and structural results, J. Mach learn. Res. 3, 463-482 (2002) · Zbl 1084.68549 · doi:10.1162/153244303321897690
[4]Cucker, F.; Smale, S.: On the mathematical foundations of learning, Bull. amer. Soc. 39, 1-49 (2001) · Zbl 0983.68162 · doi:10.1090/S0273-0979-01-00923-5
[5]Cucker, F.; Smale, S.: Best choices for regularization parameters in learning theory, Found. comput. Math. 2, 413-428 (2002) · Zbl 1057.68085 · doi:10.1007/s102080010030
[6]Douglas, R. G.: Banach algebra techniques in operator theory, (1998)
[7]Hirsch, F.; Lacombe, G.: Elements of functional analysis, (1999)
[8]Poggio, T.; Smale, S.: The mathematics of learning: dealing with data, Notices amer. Math. soc. 50, 537-544 (2003) · Zbl 1083.68100 · doi:http://www.ams.org/notices/200305/200305-toc.html
[9]Kadison, Richard V.; Ringrose, John R.: Fundamentals of the theory of operator algebras, volume 1: elementary theory, (1983)
[10]Smale, S.; Zhou, D. X.: Shannon sampling II. Connections to learning theory, Appl. comput. Harmon. anal. 19, 285-302 (2006) · Zbl 1107.94008 · doi:10.1016/j.acha.2005.03.001
[11]Smale, S.; Zhou, D. X.: Learning theory estimates via integral operators and their approximations, Constr. approx. 26, 153-172 (2007) · Zbl 1127.68088 · doi:10.1007/s00365-006-0659-y
[12]Smale, S.; Zhou, D. X.: Shannon sampling and function reconstruction from point values, Bull. amer. Math. soc. 41, 279-305 (2004) · Zbl 1107.94007 · doi:10.1090/S0273-0979-04-01025-0
[13]Sun, H. W.: Mercer theorem for RKHS on noncompact sets, J. complexity 21, 337-349 (2005) · Zbl 1094.46021 · doi:10.1016/j.jco.2004.09.002
[14]H.W. Sun, Q. Wu, Regularized least square regression with dependent samples, Adv. Comput. Math., in press (doi:10.1007/s10444-008-9099-y)
[15]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
[16]Vapnik, V.: Statistical learning theory, (1998) · Zbl 0935.62007
[17]Wu, Q.; Ying, Y. M.; Zhou, D. X.: Learning rates of least-square regularized regression, Found. comput. Math. 6, 171-192 (2006) · Zbl 1100.68100 · doi:10.1007/s10208-004-0155-9
[18]Zhou, D. X.: Capacity of reproducing kernel spaces in learning theory, IEEE trans. Inform. theory 49, 1743-1752 (2003)
[19]Zhou, D. X.: The covering number in learning theory, J. complexity 18, 739-767 (2002) · Zbl 1016.68044 · doi:10.1006/jcom.2002.0635