Kernel dimension reduction in regression

Annals of Statistics

Volumn 37, Issue 4, 2009, Pages 1871-1905

  Fukumizu, Kenji ^a   Bach, Francis R ^b   Jordan, Michael I ^c  

^a INSTITUTE OF STATISTICAL MATHEMATICS   (Japan)

^b Departement d'Informatique   (France)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Consistency; Dimension reduction; Positive definite kernel; Regression; Reproducing kernel

Indexed keywords

EID: 68649121147     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-AOS637     Document Type: Article

References (36)

1. ARONSZAJN, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 337-404. MR0051437
2. BACH, F. R. and JORDAN, M. I. (2002). Kernel independent component analysis. J. Mach. Learn. Res. 3 1-48. MR1966051
3. BAKER, C. R. (1973). Joint measures and cross-covariance operators. Trans. Amer. Math. Soc. 186 273-289. MR0336795
4. BREIMAN, L. and FRIEDMAN, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. J. Amer. Statist. Assoc. 80 580-598. MR0803258
5. CHIAROMONTE, F. and COOK, R. D. (2002). Sufficient dimension reduction and graphics in regression. Ann. Inst. Statist. Math. 54 768-795. MR1954046
6. COOK, R. D. (1998). Regression Graphics. Wiley, New York. MR1645673
7. COOK, R. D. and LEE, H. (1999). Dimension reduction in regression with a binary response. J. Amer. Statist. Assoc. 94 1187-1200. MR1731482
8. COOK, R. D. and LI, B. (2002). Dimension reduction for conditional mean in regression. Ann. Statist. 30 455-474. MR1902895
9. COOK, R. D. and WEISBERG, S. (1991). Discussion of Li. J. Amer. Statist. Assoc. 86 328-332.
10. COOK, R. D. and YIN, X. (2001). Dimension reduction and visualization in discriminant analysis (with discussion). Aust. N. Z. J. Stat. 43 147-199. MR1839361
11. FLURY, B. and RIEDWYL, H. (1988). Multivariate Statistics: A Practical Approach. Chapman and Hall, London.
12. FRIEDMAN, J. H. and STUETZLE, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823. MR0650892
13. FUKUMIZU, K., BACH, F. R. and GRETTON, A. (2007). Statistical consistency of kernel canonical correlation analysis. J. Mach. Learn. Res. 8 361-383. MR2320675
14. FUKUMIZU, K., BACH, F. R. and JORDAN, M. I. (2004). Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. J. Mach. Learn. Res. 5 73-99. MR2247974
15. FUKUMIZU, K.,GRETTON, A., SUN, X. and SCHÖLKOPF, B. (2008). Kernelmeasures of conditional dependence. In Advances in Neural Information Processing Systems 20 (J. Platt, D. Koller, Y. Singer and S. Roweis, eds.) 489-496. MIT Press, Cambridge, MA.
16. GRETTON, A., BOUSQUET, O., SMOLA, A. J. and SCHÖLKOPF, B. (2005). Measuring statistical dependence with Hilbert-Schmidt norms. In 16th International Conference on Algorithmic Learning Theory (S. Jain, H. U. Simon and E. Tomita, eds.) 63-77. Springer, Berlin. MR2255909
17. GROETSCH, C. W. (1984). The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston, MA.
18. HRISTACHE, M., JUDITSKY, A., POLZEHL, J. and SPOKOINY, V. (2001). Structure adaptive approach for dimension reduction. Ann. Statist. 29 1537-1566. MR1891738
19. KOBAYASHI, S. and NOMIZU, K. (1963). Foundations of Differential Geometry, Vol. 1.Wiley, New York.
20. LAX, P. D. (2002). Functional Analysis. Wiley, New York. MR1892228
21. LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin. MR1102015
22. LI, B., ZHA, H. and CHIAROMONTE, F. (2005). Contour regression: A general approach to dimension reduction. Ann. Statist. 33 1580-1616. MR2166556
23. LI, K.-C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316-342. MR1137117
24. LI, K.-C. (1992). On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. J. Amer. Statist. Assoc. 87 1025-1039. MR1209564
25. POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York. MR0762984
26. REED, M. and SIMON, B. (1980). Functional Analysis. Academic Press, New York. MR0751959
27. SAMAROV, A. M. (1993). Exploring regression structure using nonparametric functional estimation. J. Amer. Statist. Assoc. 88 836-847. MR1242934
28. SRIPERUMBUDUR, B.,GRETTON, A., FUKUMIZU, K., LANCKRIET, G. and SCHÖLKOPF, B. (2008). Injective Hilbert space embeddings of probability measures. In Proceedings of the 21st Annual Conference on Learning Theory (COLT 2008) (R. A. Servedio and T. Zhang, eds.) 111-122. Omnipress, Madison, WI.
29. TIBSHIRANI, R. (1996). Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58 267-288. MR1379242
30. VAKHANIA, N. N., TARIELADZE, V. I. and CHOBANYAN, S. A. (1987). Probability Distributions on Banach Spaces. Reidel, Dordrecht. MR1435288
31. VAN DER VAART, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press, Cambridge. MR1652247
32. WAHBA, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59. SIAM, Philadelphia, PA. MR1045442
33. XIA, Y., TONG, H., LI, W. and ZHU, L.-X. (2002). An adaptive estimation of dimension reduction space. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 363-410. MR1924297
34. YIN, X. and BURA, E. (2006). Moment-based dimension reduction for multivariate response regression. J. Statist. Plann. Inference 136 3675-3688. MR2256281
35. YIN, X. and COOK, R. D. (2005). Direction estimation in single-index regressions. Biometrika 92 371-384. MR2201365
36. ZHU, Y. and ZENG, P. (2006). Fourier methods for estimating the central subspace and the central mean subspace in regression. J. Amer. Statist. Assoc. 101 1638-1651. MR2279485