Rates of convex approximation in non-hilbert spaces

Springer Science and Business Media LLC - Tập 13 Số 2 - Trang 187-220 - 1997
Michael J. Donahue1, Christian J. Darken2, Leonid Gurvits3, Eduardo D. Sontag4
1Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, USA
2Learning Systems Department, Siemens Corporate Research, Inc., Princeton, USA
3NEC Research Institute, Princeton, USA
4Department of Mathematics, Rutgers University, New Brunswick, USA

Tóm tắt

Từ khóa


Tài liệu tham khảo

S. Banach, S. Saks (1930):Sur la convergence forte dans les champs L P. Studia Math.,2:51–57.

A. R. Barron (1991):Approximation and estimation bounds for artificial neural networks. In: Proc. Fourth Annual Workshop on Computational Learning Theory. Morgan Kaufmann, pp. 243–249.

A. R. Barron (1992):Neural net approximation. In: Proc. of the Seventh Yale Workshop on Adaptive and Learning Systems. pp. 69–72.

C. Bessaga, A. Pelczynski (1958):A generalization of results of R. C. James concerning absolute bases in Banach spaces. Studia Math.,17:151–164.

J. A. Clarkson (1936):Uniformly convex spaces. Trans. Amer. Math. Soc.,40:396–414.

C. Darken, M. Donahue, L. Gurvits, E. Sontag (1993):Rate of approximation results motivated by robust neural network learning. In: Proc. of the Sixth Annual ACM Conference on Computational Learning Theory. New York: The Association for Computing Machinery. pp. 303–309.

R. Deville, G. Godefroy, V. Zizler (1993): Smoothness and Renormings in Banach Spaces. New York: Wiley.

J. Dieudonné (1960): Foundations of Modern Analysis. New York: Academic Press.

T. Figiel, G. Pisier (1974):Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses. C. R. Acad. Sci. Paris,279:611–614.

W. T. Gowers (Preprint):A Banach space not containing c 0, l1, or a reflexive subspace.

U. Haagerup (1982):The best constants in the Khintchine inequality. Studia Math.,70:231–283.

O. Hanner (1956):On the uniform convexity of L p and ℓp. Arkiv. Matematik,3:239–244.

S. J. Hanson, D. J. Burr (1988):Minkowski-r back-propagation: learning in connectionist models with non-Euclidean error signals. In: Neural Information Processing Systems. New York: American Institute of Physics, p. 348.

R. C. James (1978):Nonreflexive spaces of type 2. Israel J. Math.,30:1–13.

L. K. Jones (1992):A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Ann. Statist.,20:608–613.

S. Kakutani (1938):Weak convergence in uniformly convex spaces. Tôhoku Math. J.,45:188–193.

J. Khintchine (1923):Über die diadischen Brüche. Math. Z.,18:109–116.

M. Ledoux, M. Talagrand (1991): Probability in Banach Space. Berlin: Springer-Verlag

M. Leshno, V. Lin, A. Pinkus, S. Schocken (1992):Multilayer feedforward networks with a non-polynomial activation function can approximate any function. Preprint. Hebrew University.

J. Lindenstrauss (1963):On the modulus of smoothness and divergent series in Banach spaces. Michigan Math. J.,10:241–252.

J. Lindenstrauss, L. Tzafriri (1979): Classical Banach Spaces II: Function Spaces. Berlin: Springer-Verlag.

M. J. D. Powell (1981): Approximation Theory and Methods. Cambridge: Cambridge University Press.

W. J. Rey (1983): Introduction to Robust and Quasi-Robust Statistical Methods. Berlin: Springer-Verlag.

H. Rosenthal (1974):A characterization of Banach spaces containing l l. Proc. Nat. Acad. Sci. (USA),71:2411–2413.

H. Rosenthal (1994):A subsequence principle characterizing Banach spaces containing c 0. Bull. Amer. Math. Soc.,30:227–233.

E. D. Sontag (1992):Feedback stabilization using two-hidden-layer nets. IEEE Trans. Neural Networks,3:981–990.

K. R. Stromberg (1981): An Introduction to Classical Real Analysis. New York: Wadsworth.

J. Y. T. Woo (1973):On modular sequence spaces. Studia Math.,48:271–289.