jmlr jmlr2005 jmlr2005-35 jmlr2005-35-reference knowledge-graph by maker-knowledge-mining
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Author: Alain Rakotomamonjy, Stéphane Canu
Abstract: This work deals with a method for building a reproducing kernel Hilbert space (RKHS) from a Hilbert space with frame elements having special properties. Conditions on existence and a method of construction are given. Then, these RKHS are used within the framework of regularization theory for function approximation. Implications on semiparametric estimation are discussed and a multiscale scheme of regularization is also proposed. Results on toy and real-world approximation problems illustrate the effectiveness of such methods. Keywords: regularization, kernel, frames, wavelets
U. Amato, A. Antoniadis, and M. Pensky. Wavelet kernel penalized estimation for non-equispaced design regression. Statistics and Computing, to appear, 2004. N. Aronszajn. Theory of reproducing kernels. Trans. Am. Math. Soc., (68):337–404, 1950. M. Atteia. Hilbertian kernels and spline functions. North-Holland, 1992. M. Atteia and J. Gaches. Approximation hilbertienne : Splines, Ondelettes, Fractales. Presses Universitaires de Grenoble, 1999. A. Berlinet and C. Thomas Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers, 2004. B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, 5th Annual ACM Workshop on COLT, pages 144–152, Pittsburgh, PA, 1992. ACM Press. H. Brezis. Analyse fonctionnelle, Th´ orie et applications. Masson, 1983. e C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2), 1998. O. Christensen. Frame decomposition in Hilbert Spaces. PhD thesis, Aarhus Univ. Danemmark and Univ. of Vienna, Austria, 1993. I. Daubechies. Ten Lectures on Wavelet. SIAM, CBMS-NSF regional conferences edition, 1992. L. Debnath and P. Mikusinki. Introduction to Hilbert Spaces with applications. Academic Press, 1998. R. Duffin and A. Schaeffer. A class of nonharmonic fourier series. Trans. Amer. Math. Soc., 72: 341–366, 1952. T. Evgeniou, M. Pontil, and T. Poggio. Regularization Networks and Support Vector Machines. Advances in Computational Mathematics, 13(1):1–50, 2000. J. Gao, C. Harris, and S. Gunn. On a class of a support vector kernels based on frames in function hilbert spaces. Neural Computation, 13(9):1975–1994, 2001. F. Girosi. An equivalence between sparse approximation and support vector machines. Neural Computation, 10(6):1455–1480, 1998. F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural Computation, 7(2):219–269, 1995. K. Grochenig. Acceleration of the frame algorithm. IEEE Trans. Signal Proc., 41(12):3331–3340, 1993. C. Groetsch. Inverse Problems in the mathematical sciences. Vieweg and Sohn, 1993. 1514 F RAMES , R EPRODUCING K ERNELS , R EGULARIZATION AND L EARNING T. Hastie and R. Tibshirani. Generalized additive models. Chapman and Hall, 1990. R. Hyndman and M. Akram. Time Series data Library. Technical report, University of Monash, Dept of Econometrics and Business Statistics, 1998. http://wwwpersonal.buseco.monash.edu.au/ hyndman/TSDL/index.htm. T. Jaakkola and D. Haussler. Probabilistic kernel regression models. In Proceedings of the 1999 Conference on AI and Statistics, 1999. G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. J. Math. Anal. Applic., 33:82–95, 1971. S. Mallat. A wavelet tour of signal processing. Academic Press, 1998. V. Morosov. Methods for solving incorrectly posed problems. Springer Verlag, 1984. P. Niyogi, F. Girosi, and T. Poggio. Incorporating prior information in machine learning by creating virtual examples. In Proceedings of the IEEE, volume 86, pages 2196–2209, 1998. R. Opfer. Multiscale kernels. Technical report, Institut fur Numerische und Angewandte Mathematik, Universitt Gottingen, 2004a. R. Opfer. Tight frame expansions of multiscale reproducing kernels in Sobolev spaces. Technical report, Institut fur Numerische und Angewandte Mathematik, Universitt Gottingen, 2004b. B. Scholkopf, P. Y. Simard, A. J. Smola, and V. Vapnik. Prior knowledge in support vector kernels. In M. I. Jordan, M. J. Kearns, and S. A. Solla, editors, Advances in Neural information processings systems, volume 10, pages 640–646, Cambridge, MA, 1998. MIT Press. A. Smola. Learning with Kernels. PhD thesis, Published by: GMD, Birlinghoven, 1998. A. Smola, B. Scholkopf, and KR Muller. The connection between regularization operators and support vector kernels. Neural Networks, 11:637–649, 1998. A. Tikhonov and V. Ars´ nin. Solutions of Ill-posed problems. W.H. Winston, 1977. e V. Vapnik. The Nature of Statistical Learning Theory. Springer, N.Y, 1995. V. Vapnik. Statistical Learning Theory. Wiley, 1998. V. Vapnik, S. Golowich, and A. Smola. Support Vector Method for function estimation, Regression estimation and Signal processing, volume Vol. 9. MIT Press, Cambridge, MA, neural information processing systems, edition, 1997. G. Wahba. Spline Models for Observational Data. Series in Applied Mathematics, Vol. 59, SIAM, 1990. G. Wahba. An introduction to model building with reproducing kernel hilbert spaces. Technical Report TR-1020, University of Wisconsin-Madison, 2000. 1515