Equivalence of distance-based and RKHS-based statistics in hypothesis testing

Annals of Statistics

Volumn 41, Issue 5, 2013, Pages 2263-2291

  Sejdinovic, Dino ^a   Sriperumbudur, Bharath ^b   Gretton, Arthur ^a   Fukumizu, Kenji ^c  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c INSTITUTE OF STATISTICAL MATHEMATICS   (Japan)

Author keywords

Distance covariance; Independence testing; Reproducing kernel Hilbert spaces; Two sample testing

Indexed keywords

EID: 84888856531     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/13-AOS1140     Document Type: Article

References (36)

1. ALBA FERNÁNDEZ, V., JIMÉNEZ GAMERO, M. D. andMUÑOZ GARCÍA, J. (2008). A test for the two-sample problem based on empirical characteristic functions. Comput. Statist. Data Anal. 52 3730-3748. MR2427377
2. ANDERSON, N. H., HALL, P. and TITTERINGTON, D. M. (1994). Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernelbased density estimates. J. Multivariate Anal. 50 41-54. MR1292607
3. ARCONES, M. A. and GINÉ, E. (1992). On the bootstrap of U and V statistics. Ann. Statist. 20 655-674. MR1165586
4. BACH, F. R. and JORDAN, M. I. (2002). Kernel independent component analysis. J. Mach. Learn. Res. 3 1-48. MR1966051
5. BARINGHAUS, L. and FRANZ, C. (2004). On a new multivariate two-sample test. J. Multivariate Anal. 88 190-206. MR2021870
6. BERG, C., CHRISTENSEN, J. P. R. and RESSEL, P. (1984). Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics 100. Springer, New York. MR0747302
7. BERLINET, A. and THOMAS-AGNAN, C. (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, London.
8. FUKUMIZU, K., SONG, L. and GRETTON, A. (2011). Kernel Bayes' rule. In Advances in Neural Information Processing Systems (J. Shawe-Taylor, R. S. Zemel, P. Bartlett, F. C. N. Pereira and K. Q. Weinberger, eds.) 24 1737-1745. Curran Associates, Red Hook, NY.
9. FUKUMIZU, K., GRETTON, A., SUN, X. and SCHÖLKOPF, B. (2008). Kernel measures of conditional dependence. In Advances in Neural Information Processing Systems 20 489-496. MIT Press, Cambridge, MA.
10. FUKUMIZU, K., SRIPERUMBUDUR, B.,GRETTON, A. and SCHOELKOPF, B. (2009). Characteristic kernels on groups and semigroups. In Advances in Neural Information Processing Systems 21 473-480. Curran Associates, Red Hook, NY.
11. GRETTON, A., FUKUMIZU, K. and SRIPERUMBUDUR, B. K. (2009). Discussion of: Brownian distance covariance. Ann. Appl. Stat. 3 1285-1294. MR2752132
12. GRETTON, A. and GYÖRFI, L. (2010). Consistent nonparametric tests of independence. J. Mach. Learn. Res. 11 1391-1423. MR2645456
13. GRETTON, A., BOUSQUET, O., SMOLA, A. and SCHÖLKOPF, B. (2005). Measuring statistical dependence with Hilbert-Schmidt norms. In Algorithmic Learning Theory (S. Jain, H. U. Simon and E. Tomita, eds.). Lecture Notes in Computer Science 3734 63-77. Springer, Berlin. MR2255909
14. GRETTON, A., BORGWARDT, K., RASCH, M., SCHÖLKOPF, B. and SMOLA, A. (2007). A kernel method for the two-sample problem. In NIPS 513-520. MIT Press, Cambridge, MA.
15. GRETTON, A., FUKUMIZU, K., TEO, C. H., SONG, L., SCHÖLKOPF, B. and SMOLA, A. (2008). A kernel statistical test of independence. In Advances in Neural Information Processing Systems 20 585-592. MIT Press, Cambridge, MA.
16. GRETTON, A., FUKUMIZU, K., HARCHAOUI, Z. and SRIPERUMBUDUR, B. (2009). A fast, consistent kernel two-sample test. In Advances in Neural Information Processing Systems 22. Curran Associates, Red Hook, NY.
17. GRETTON, A., BORGWARDT, K. M., RASCH, M. J., SCHÖLKOPF, B. and SMOLA, A. (2012a). A kernel two-sample test. J. Mach. Learn. Res. 13 723-773. MR2913716
18. GRETTON, A., SRIPERUMBUDUR, B., SEJDINOVIC, D., STRATHMANN, H., BALAKRISHNAN, S., PONTIL, M. and FUKUMIZU, K. (2012b). Optimal kernel choice for large-scale two-sample tests. In Advances in Neural Information Processing Systems 25 1214-1222. Curran Associates, Red Hook, NY.
19. LYONS, R. (2013). Distance covariance in metric spaces. Ann. Probab. 41 3284-3305.
20. MÜLLER, A. (1997). Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29 429-443. MR1450938
21. REED, M. and SIMON, B. (1980). Methods of Modern Mathematical Physics. I: Functional Analysis, 2nd ed. Academic Press, San Diego. MR0751959
22. SCHÖLKOPF, B., SMOLA, A. J. and MÜLLER, K. R. (1997). Kernel principal component analysis. In ICANN (W. Gerstner, A. Germond, M. Hasler and J. D. Nicoud, eds.). Lecture Notes in Computer Science 1327 583-588. Springer, Berlin.
23. SEJDINOVIC, D., GRETTON, A., SRIPERUMBUDUR, B. and FUKUMIZU, K. (2012). Hypothesis testing using pairwise distances and associated kernels. In Proceedings of the International Conference on Machine Learning (ICML) 1111-1118. Omnipress, New York.
24. SMOLA, A. J., GRETTON, A., SONG, L. and SCHÖLKOPF, B. (2007). A Hilbert space embedding for distributions. In Proceedings of the Conference on Algorithmic Learning Theory (ALT) 4754 13-31. Springer, Berlin.
25. SRIPERUMBUDUR, B. (2011). Mixture density estimation via Hilbert space embedding of measures. In Proceedings of the International Symposium on Information Theory 1027-1030. IEEE, Piscataway, NJ.
26. SRIPERUMBUDUR, B. K., FUKUMIZU, K. and LANCKRIET, G. R. G. (2011). Universality, characteristic kernels and RKHS embedding of measures. J. Mach. Learn. Res. 12 2389-2410. MR2825431
27. SRIPERUMBUDUR, B., GRETTON, A., FUKUMIZU, K., LANCKRIET, G. and SCHÖLKOPF, B. (2008). Injective Hilbert space embeddings of probability measures. In Proceedings of the Conference on Learning Theory (COLT) 111-122. Omnipress, New York.
28. SRIPERUMBUDUR, B., FUKUMIZU, K., GRETTON, A., LANCKRIET, G. and SCHOELKOPF, B. (2009). Kernel choice and classifiability for RKHS embeddings of probability distributions. In Advances in Neural Information Processing Systems 22. Curran Associates, Red Hook, NY.
29. SRIPERUMBUDUR, B. K., GRETTON, A., FUKUMIZU, K., SCHÖLKOPF, B. and LANCKRIET, G. R. G. (2010). Hilbert space embeddings and metrics on probability measures. J. Mach. Learn. Res. 11 1517-1561. MR2645460
30. SRIPERUMBUDUR, B. K., FUKUMIZU, K., GRETTON, A., SCHÖLKOPF, B. and LANCKRIET, G. R. G. (2012). On the empirical estimation of integral probability metrics. Electron. J. Stat. 6 1550-1599. MR2988458
31. STEINWART, I. and CHRISTMANN, A. (2008). Support Vector Machines. Springer, New York. MR2450103
32. SZÉKELY, G. and RIZZO, M. (2004). Testing for equal distributions in high dimension. InterStat 5.
33. SZÉKELY, G. J. and RIZZO, M. L. (2005). A new test for multivariate normality. J. Multivariate Anal. 93 58-80. MR2119764
34. SZÉKELY, G. J., RIZZO, M. L. and BAKIROV, N. K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769-2794. MR2382665
35. SZÉKELY, G. J. and RIZZO, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236- 1265. MR2752127
36. ZHANG, K., PETERS, J., JANZING, D. and SCHOELKOPF, B. (2011). Kernel-based conditional independence test and application in causal discovery. In Proceedings of the Conference on Uncertainty in Artificial Intelligence (UAI) 804-813. AUAI Press, Corvallis, Oregon.