Approximation and learning by greedy algorithms

Annals of Statistics

Volumn 36, Issue 1, 2008, Pages 64-94

  Barron, Andrew R ^a   Cohen, Albert ^b,e   Dahmen, Wolfgang ^c   DeVore, Ronald A ^d  

^a YALE UNIVERSITY   (United States)

^b SORBONNE UNIVERSITÉ   (France)

^c RWTH AACHEN UNIVERSITY   (Germany)

^d UNIVERSITY OF SOUTH CAROLINA   (United States)

^e UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Convergence rates for greedy algorithms; Interpolation spaces; Neural networks; Nonparametric regression; Statistical learning

Indexed keywords

EID: 50649107089     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053607000000631     Document Type: Article

References (23)

1. AVELLANEDA, M., DAVIS, G. and MALLAT, S. (1997). Adaptive greedy approximations. Constr. Approx. 13 57-98. MR1424364
2. BARRON, A. R. (1990). Complexity regularization with application to artificial neural network. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 561-576. Kluwer Academic Publishers, Dordrecht. MR1154352
3. BARRON, A. R. (1992). Neural net approximation. Proc. 7th Yale Workshop on Adaptive and Learning Systems (K. S. Narendra, ed.) 1 69-72. New Haven, CT.
4. BARRON, A. R. (1993). Universal approximation bounds for superposition of n sigmoidal functions. IEEE Trans. Inform. Theory 39 930-945. MR 1237720
5. BARRON, A. and CHEANG, G. H. L. (2001). Penalized least squares, model selection, convex hull classes. and neural nets. In Proceedings of the 9th ESANN, Brugge. Belgium (M. Verleysen, ed.) 371-376. De-Facto Press.
6. BENNETT, C. and SHARPLEY, R. (1988). Interpolation of Operators. Academic Press, Boston. MR0928802
7. BERGH, J. and LÖFSTRÖM, J. (1976). Interpolation Spaces. Springer, Berlin. MR0482275
8. DEVORE, R. (1998). Nonlinear approximation. In Acta Numerica 7 51-150. Cambridge Univ. Press. MR1689432
9. DEVORE, R. and TEMLYAKOV, V. (1996). Some remarks on greedy algorithms. Adu Comput. Math. 5 173-187. MR 1399379
10. EFRON, B., HASTIE, T., JOHNSTONE. I. and TIBSHIRANI, R. (2004). Least angle regression (with discussion). Ann. Statist. 32 407-499. MR2060166
11. GYÖRFY, L., KOHLER, M., KRZYZAK, A. and WALK, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer, Berlin. MR1920390
12. HASTIE, T., TIBSHIRANI, R. and FRIEDMAN, J. (2001). The Elements of Statistical Learning. Springer, New York. MR1851606
13. 1- penalized estimators from flexible function libraries. Yale Department of Statistics Report.
14. JONES, L. K. (1992). A simple lemma on greedy approximation in Hilbert spaces and convergence rates for projection pursuit regression and neural network training. Ann. Statist. 20 608-613. MR1150368
15. KONYAGIN, S. V. and TEMLYAKOV, V. N. (1999). Rate of convergence of pure greedy algorithm. East J. Approx. 5 493-499. MR 1738484
16. KURKOVA, V. and SANGUINETI, M. (2001). Bounds on rates of variable-basis and neuralnetwork approximation. IEEE Trans. Inform. Theory 47 2659-2665. MR 1873958
17. KURKOVA, V. and SANGUINETI, M. (2002). Comparison of worst case errors in linear and neural network approximation. IEEE Trans. Inform. Theory 48 264-275. MR1872179
18. LEE, W. S., BARTLETT, P. and WILLIAMSON, R. C. (1996). Efficient agnostic learning of neural networks with bounded fan-in. IEEE Trans. Inform. Theory 42 2118-2132. MR1447518
19. LIVSHITZ, E. D. and TEMLYAKOV, V. N. (2003). Two lower estimates in greedy approximation. Constr. Approx. 19 509-524. MR1998902
20. PETRUSHEV, P. P. (1998). Approximation by ridge functions and neural networks. SIAM J. Math. Anal. 30 155-189. MR 1646689
21. TEMLYAKOV, V. (2003). Nonlinear methods of approximation. J. FOCM 3 33-107. MR1951502
22. TEMLYAKOV. V. (2005). Greedy algorithms with restricted depth search. Proc. of the Steklov Inst. Math. 248 255-267. MR2165933
23. TIBSHIRANI, R. (1995). Regression shrinkage and selection via the LASSO. J. Roy. Statist. Soc. Ser. B 58 267-288. MR 1379242