Bulletin of the American Mathematical Society

Author(s): Felipe Cucker; Steve Smale
Journal: Bull. Amer. Math. Soc. 39 (2002), 1-49.
MSC (2000): Primary 68T05, 68P30
Posted: October 5, 2001
Retrieve article in: PDF

1.: L.V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill, 1978. MR 80c:30001
2.: N. Aronszajn, Theory of reproducing kernels, Transactions of the Amer. Math. Soc.68 (1950), 337-404. MR 14:479c
3.: A.R. Barron, Complexity regularization with applications to artificial neural networks, Nonparametric Functional Estimation (G. Roussa, ed.), Kluwer, Dordrecht, 1990, pp. 561-576. MR 93b:62052
4.: J. Bergh and J. Löfström, Interpolation spaces. an introduction, Springer-Verlag, 1976. MR 58:2349
5.: M.S. Birman and M.Z. Solomyak, Piecewise polynomial approximations of functions of the classes ${W}^\alpha_p$ , Mat. Sb. 73 (1967), 331-355; English translation in Math. USSR Sb. (1967), 295-317. MR 36:576
6.: C.M. Bishop, Neural networks for pattern recognition, Cambridge University Press, 1995. MR 97m:68172
7.: A. Björck, Numerical methods for least squares problems, SIAM, 1996. MR 97g:65004
8.: L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and real computation, Springer-Verlag, 1998. MR 99a:68070
9.: B. Carl and I. Stephani, Entropy, compactness and the approximation of operators, Cambridge University Press, 1990. MR 92e:47002
10.: P. Craven and G. Wahba, Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. Math. 31 (1979), 377-403. MR 81g:65018
11.: C. Darken, M. Donahue, L. Gurvits, and E. Sontag, Rates of convex approximation in non-Hilbert spaces, Construct. Approx. 13 (1997), 187-220. MR 98c:41051
12.: L. Debnath and P. Mikusinski, Introduction to Hilbert spaces with applications, 2nd ed., Academic Press, 1999. MR 99k:46001
13.: J.-P. Dedieu and M. Shub, Newton's method for overdetermined systems of equations, Mathematics of Computation 69 (2000), 1099-1115. MR 2000j:65133
14.: R.A. DeVore and G.G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften, vol. 303, Springer-Verlag, 1993. MR 95f:41001
15.: J. Duchon, Spline minimizing rotation-invariant semi-norms in Sobolev spaces, Constructive theory of functions on several variables (W. Schempp and K. Zeller, eds.), Lecture Notes in Math. 571, Springer-Verlag, Berlin, 1977. MR 58:12146
16.: D.E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge University Press, 1996. MR 97h:46045
17.: T. Evgeniou, M. Pontil, and T. Poggio, Regularization Networks and Support Vector Machines, Advances in Computational Mathematics 13 (2000), 1-50. MR 2001f:68053
18.: D. Haussler, Decision theoretic generalizations of the PAC model for neural net and other learning applications, Information and Computation 100 (1992), 78-150. MR 93i:68149
19.: H. Hochstadt, Integral equations, John Wiley & Sons, 1973. MR 52:11503
20.: A.N. Kolmogorov and S.V. Fomin, Introductory real analysis, Dover Publications Inc., 1975. MR 51:13617
21.: A.N. Kolmogorov and V.M. Tikhomirov, $\varepsilon$ -entropy and $\varepsilon$ -capacity of sets in function spaces, Uspecki 14 (1959), 3-86. MR 22:2890
22.: W.-S. Lee, P. Bartlett, and R. Williamson, The importance of convexity in learning with squared loss, IEEE Transactions on Information Theory 44 (1998), 1974-1980. MR 99k:68160
23.: P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. MR 87f:58156
24.: G.G. Lorentz, M. Golitschek, and Y. Makovoz, Constructive approximation; advanced problems, Springer-Verlag, 1996. MR 97k:41002
25.: W.S. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics 5 (1943), 115-133. MR 6:12a
26.: J. Meinguet, Multivariate interpolation at arbitrary points made simple, J. Appl. Math. Phys. 30 (1979), 292-304. MR 81e:41014
27.: M.L. Minsky and S.A. Papert, Perceptrons, MIT Press, 1969.
28.: P. Niyogi, The informational complexity of learning, Kluwer Academic Publishers, 1998.
29.: A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, 1987. MR 88j:47022b
30.: A. Pinkus, -widths in approximation theory, Springer-Verlag, New York, 1986. MR 86k:41001
31.: T. Poggio and C.R. Shelton, Machine learning, machine vision, and the brain, AI Magazine 20 (1999), 37-55.
32.: D. Pollard, Convergence of stochastic processes, Springer-Verlag, 1984. MR 86i:60074
33.: G.V. Rozenblum, M.A. Shubin, and M.Z. Solomyak, Partial differential equations vii: Spectral theory of differential operators, Encyclopaedia of Mathematical Sciences, vol. 64, Springer-Verlag, 1994.
34.: I.J. Schoenberg, Metric spaces and completely monotone functions, Ann. of Math. 39 (1938), 811-841.
35.: I.R. Shafarevich, Basic algebraic geometry, 2nd ed., vol. 1: Varieties in Projective Space, Springer-Verlag, 1994. MR 95m:14001
36.: S. Smale, On the Morse index theorem, J. Math. and Mech. 14 (1965), 1049-1056, With a Corrigendum in J. Math. and Mech. 16, 1069-1070, (1967). MR 31:6251; MR 34:5108
37.: -, Mathematical problems for the next century, Mathematics: Frontiers and Perspectives (V. Arnold, M. Atiyah, P. Lax, and B. Mazur, eds.), AMS, 2000, pp. 271-294. CMP 2000:13
38.: S. Smale and D.-X. Zhou, Estimating the approximation error in learning theory, Preprint, 2001.
39.: M.E. Taylor, Partial differential equations i: Basic theory, Applied Mathematical Sciences, vol. 115, Springer-Verlag, 1996. MR 98b:35002b
40.: L.G. Valiant, A theory of the learnable, Communications of the ACM27 (1984), 1134-1142.
41.: S. van de Geer, Empirical processes in m-estimation, Cambridge University Press, 2000.
42.: V. Vapnik, Statistical learning theory, John Wiley & Sons, 1998. MR 99h:62052
43.: P. Venuvinod, Intelligent production machines: benefiting from synergy amongst modelling, sensing and learning, Intelligent Production Machines: Myths and Realities, CRC Press LLC, 2000, pp. 215-252.
44.: A.G. Vitushkin, Estimation of the complexity of the tabulation problem, Nauka (in Russian), 1959, English Translation appeared as Theory of the Transmission and Processing of the Information, Pergamon Press, 1961.
45.: G. Wahba, Spline models for observational data, SIAM, 1990. MR 91g:62028
46.: R. Williamson, A. Smola, and B. Schölkopf, Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators, Tech. Report NC2-TR-1998-019, NeuroCOLT2, 1998.

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 68T05, 68P30

Felipe Cucker
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Email: macucker@math.cityu.edu.hk

Steve Smale
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Address at time of publication: Department of Mathematics, University of California, Berkeley, California 94720
Email: masmale@math.cityu.edu.hk, smale@math.berkeley.edu

DOI: 10.1090/S0273-0979-01-00923-5
PII: S 0273-0979(01)00923-5
Received by editor(s): April 2000, and in revised form June 1, 2001
Posted: October 5, 2001
Additional Notes: This work has been substantially funded by CERG grant No. 9040457 and City University grant No. 8780043.
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS