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A note on application of integral operator in learning theory. (English) Zbl 1165.68059
Summary: By the aid of the properties of the square root of positive operators we refine the consistency analysis of regularized least square regression in a reproducing kernel Hilbert space. Sharper error bounds and faster learning rates are obtained when the sampling sequence satisfies a strongly mixing condition.
MSC:
68T05Learning and adaptive systems
References:
[1]Aronszajn, N.: Theory of reproducing kernels, Trans. amer. Math. soc. 68, 337-404 (1950) · Zbl 0037.20701 · doi:10.2307/1990404
[2]Athreya, K. B.; Pantula, S. G.: Mixing properties of harris chains and autoregressive processes, J. appl. Probab. 23, 880-892 (1986) · Zbl 0623.60087 · doi:10.2307/3214462
[3]Bousquet, O.; Elisseeff, A.: Stability and generalization, J. Mach learn. Res. 2, 499-526 (2002) · Zbl 1007.68083 · doi:10.1162/153244302760200704
[4]Carrasco, M.; Chen, X.: Mixing and moment properties of various GARCH and stochastic volatility models, Econom. theory 18, 17-39 (2002) · Zbl 1181.62125 · doi:10.1017/S0266466602181023
[5]Cucker, F.; Zhou, D. X.: Learning theory: an approximation theory viewpoint, (2007)
[6]Douglas, R. G.: Banach algebra techniques in operator theory, (1998)
[7]Evgeniou, T.; Pontil, M.; Poggio, T.: Regularization networks and support vector machines, Adv. comput. Math. 13, 1-50 (2000) · Zbl 0939.68098 · doi:10.1023/A:1018946025316
[8]Gouriéroux, C.: ARCH model and financial application, (1997)
[9]Lowner, K.: Über monotone matrixfunktionen, Math. Z. 38, 177-216 (1934) · Zbl 0008.11301 · doi:10.1007/BF01170633
[10]Modha, D. S.: Minimum complexity regression estimation with weakly dependent observations, IEEE trans. Inform. theory 42, 2133-2145 (1996) · Zbl 0868.62015 · doi:10.1109/18.556602
[11]Pedersen, G. K.: Some operator monotone functions, Proc. amer. Math. soc. 36, 309-310 (1972) · Zbl 0256.47019 · doi:10.2307/2039083
[12]Smale, S.; Zhou, D. X.: Estimating the approximation error in learning theory, Anal. appl. 1, 17C41 (2003) · Zbl 1079.68089 · doi:10.1142/S0219530503000089
[13]Smale, S.; Zhou, D. X.: Shannon sampling II: Connections to learning theory, Appl. comput. Harmon. anal. 19, 285-302 (2005) · Zbl 1107.94008 · doi:10.1016/j.acha.2005.03.001
[14]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
[15]S. Smale, D.X. Zhou, Online learning with Markov sampling, Anal. Appl., in press · Zbl 1170.68022 · doi:10.1142/S0219530509001293
[16]H.W. Sun, Q. Wu, Regularized least square regression with dependent samples, Adv. Comput. Math. (2008), doi:10.1007/s10444-008-9099-y, in press
[17]Vapnik, V. N.: Statistical learning theory, (1998) · Zbl 0935.62007
[18]White, H.: Connectionist nonparametric regression: multilayer feedforward networks can learn arbitrary mappings, Neural networks 3, 535-549 (1990)
[19]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
[20]Xu, Y. L.; Chen, D. R.: Learning rates of regularized regression for exponentially strongly mixing sequence, J. statist. Plann. 138, No. 7, 2180-2189 (2008) · Zbl 1134.62050 · doi:10.1016/j.jspi.2007.09.003
[21]Zhang, T.: Leave-one-out bounds for kernel methods, Neural comput. 15, 1397-1437 (2003) · Zbl 1085.68144 · doi:10.1162/089976603321780326