Mercer's Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Constructive Approximation

Volumn 35, Issue 3, 2012, Pages 363-417

  Steinwart, Ingo ^a   Scovel, Clint ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Eigenvalues; Integral operators; Interpolation spaces; Reproducing Kernel Hilbert spaces; Statistical learning theory

Indexed keywords

EID: 84860443821     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-012-9153-3     Document Type: Article

References (47)

1. Agler, J., McCarthy, J. E.: Pick Interpolation and Hilbert Function Spaces. Am. Math. Soc., Providence (2002).
2. Ali, S. T., Antoine, J.-P., Gazeau, J.-P.: Coherent States, Wavelets and Their Generalizations. Springer, New York (2000).
3. Alpay, D. (ed.): Reproducing Kernel Spaces and Applications. Birkhäuser Verlag, Basel (2003).
4. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337-404 (1950).
5. Bauer, H.: Measure and Integration Theory. De Gruyter, Berlin (2001).
6. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988).
7. Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, Boston (2004).
8. Caponnetto, A., De Vito, E.: Fast rates for regularized least-squares algorithm. Technical Report CBCL Paper #248, AI Memo #2005-013, MIT, Cambridge, MA (2005).
9. Caponnetto, A., De Vito, E.: Optimal rates for regularized least squares algorithm. Found. Comput. Math. 7, 331-368 (2007).
10. Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, Cambridge (2006).
11. Conway, J. B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990).
12. Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines. Cambridge University Press, Cambridge (2000).
13. Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Am. Math. Soc. 39, 1-49 (2002).
14. Cucker, F., Zhou, D. X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007).
15. De Vito, E., Caponnetto, A., Rosasco, L.: Model selection for regularized least-squares algorithm in learning theory. Found. Comput. Math. 5, 59-85 (2005).
16. Hein, M., Bousquet, O.: Kernels, associated structures and generalizations. Technical Report 127, Max-Planck-Institute for Biological Cybernetics (2004).
17. Hille, E.: Introduction to general theory of reproducing kernels. Rocky Mt. J. Math. 2, 321-368 (1972).
18. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin-New York (1976).
19. König, H.: Eigenvalue Distribution of Compact Operators. Birkhäuser, Basel (1986).
20. Mendelson, S., Neeman, J.: Regularization in kernel learning. Ann. Stat. 38, 526-565 (2010).
21. Meschkowski, H.: Hilbertsche Räume mit Kernfunktion. Springer, Berlin (1962).
22. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Linear Information, vol. 1. European Mathematical Society (EMS), Zürich (2008).
23. Pietsch, A.: Eigenvalues and s-Numbers. Geest & Portig K.-G., Leipzig (1987).
24. Rao, M. M.: Measure Theory and Integration, 2nd edn. Dekker, New York (2004).
25. Riesz, F., Nagy, B. Sz.: Functional Analysis, 2nd edn. Dover, New York (1990).
26. Ritter, K.: Average-Case Analysis of Numerical Problems. Lecture Notes in Math., vol. 1733. Springer, Berlin (2000).
27. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991).
28. Saitoh, S.: Theory of Reproducing Kernels and Applications. Longman Scientific & Technical, Harlow (1988).
29. Saitoh, S.: Integral Transforms, Reproducing Kernels and Their Applications. Longman Scientific & Technical, Harlow (1997).
30. Saitoh, S., Alpay, D., Ball, J. A., Ohsawa, T. (eds.): Reproducing Kernels and Their Applications, Kluwer Academic, Dordrecht (1999).
31. Schölkopf, B., Smola, A. J.: Learning with Kernels. MIT Press, Cambridge (2002).
32. Schölkopf, B., Smola, A. J., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299-1319 (1998).
33. Shawe-Taylor, J., Williams, C. K. I., Cristianini, N., Kandola, J.: On the eigenspectrum of the Gram matrix and the generalization error of kernel-PCA. IEEE Trans. Inf. Theory 51, 2510-2522 (2005).
34. Smale, S., Zhou, D.-X.: Estimating the approximation error in learning theory. Anal. Appl. 1, 17-41 (2003).
35. Smale, S., Zhou, D.-X.: Learning theory estimates via integral operators and their approximations. Constr. Approx. 26, 153-172 (2007).
36. Steinwart, I., Christmann, A.: Support Vector Machines. Springer, New York (2008).
37. Steinwart, I., Hush, D., Scovel, C.: Optimal rates for regularized least squares regression. In: Dasgupta, S., Klivans, A. (eds.) Proceedings of the 22nd Annual Conference on Learning Theory, pp. 79-93 (2009).
38. Steinwart, I., Hush, D., Scovel, C.: Training SVMs without offset. J. Mach. Learn. Res. 12, 141-202 (2011).
39. Strauss, W., Macheras, N. D., Musiał, K.: Liftings. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1131-1184. Elsevier, Amsterdam (2002).
40. Sun, H.: Mercer theorem for RKHS on noncompact sets. J. Complex. 21, 337-349 (2005).
41. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007).
42. Ionescu Tulcea, A., Ionescu Tulcea, C.: Topics in the Theory of Lifting. Springer, New York (1969).
43. Wahba, G.: Spline Models for Observational Data. Series in Applied Mathematics, vol. 59. SIAM, Philadelphia (1990).
44. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005).
45. Werner, D.: Funktionalanalysis. Springer, Berlin (1995).
46. Yao, Y., Rosasco, L., Caponnetto, A.: On early stopping in gradient descent learning. Constr. Approx. 26, 289-315 (2007).
47. Zhou, D.-X.: The covering number in learning theory. J. Complex. 18, 739-767 (2002).