Two-stage sampled learning theory on distributions

Journal of Machine Learning Research

Volumn 38, Issue , 2015, Pages 948-957

  Szabó, Zoltán ^a   Gretton, Arthur ^a   Póczos, Barnabás ^b   Sriperumbudur, Bharath ^c  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c PSU   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARTIFICIAL INTELLIGENCE; REGRESSION ANALYSIS;

GENERALIZATION PERFORMANCE; KERNEL DENSITY ESTIMATION; PROBLEM DIFFICULTY; REGRESSION PROBLEM; REPRODUCING KERNEL HILBERT SPACES; SIMILARITY MEASURE; SPECIFIC LEARNING; THEORETICAL GUARANTEES;

PROBABILITY DISTRIBUTIONS;

EID: 84954309811     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (43)

1. Barnabás Póczos, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman. Distributionfree distribution regression. AISTATS (JMLR W&CP), 31: 507-515, 2013.
2. Thomas G. Dietterich, Richard H. Lathrop, and Tomás Lozano-Pérez. Solving the multiple instance problem with axis-parallel rectangles. Artificial Intelligence, 89: 31-71, 1997.
3. Soumya Ray and David Page. Multiple instance regression. In ICML, pages 425-432, 2001.
4. Daniel R. Dooly, Qi Zhang, Sally A. Goldman, and Robert A. Amar. Multiple-instance learning of real-valued data. Journal of Machine Learning Research, 3: 651-678, 2002.
5. Andrea Caponnetto and Ernesto De Vito. Optimal rates for regularized least-squares algorithm. Foundations of Computational Mathematics, 7: 331-368, 2007.
6. Ingo Steinwart and Andres Christmann. Support Vector Machines. Springer, 2008.
7. David Haussler. Convolution kernels on discrete structures. Technical report, Department of Computer Science, University of California at Santa Cruz, 1999. (http://cbse.soe.ucsc.edu/ sites/default/files/convolutions.pdf).
8. Fei Wang, Tanveer Syeda-Mahmood, Baba C. Vemuri, David Beymer, and Anand Rangarajan. Closed-form Jensen-Rényi divergence for mixture of Gaussians and applications to groupwise shape registration. Medical Image Computing and Computer-Assisted Intervention, 12: 648-655, 2009.
9. Risi Kondor and Tony Jebara. A kernel between sets of vectors. In ICML, pages 361-368, 2003.
10. Tony Jebara, Risi Kondor, and Andrew Howard. Probability product kernels. Journal of Machine Learning Research, 5: 819-844, 2004.
11. Frank Nielsen and Richard Nock. A closed-form expression for the Sharma-Mittal entropy of exponential families. Journal of Physics A: Mathematical and Theoretical, 45: 032003, 2012.
12. Shaohua Kevin Zhou and Rama Chellappa. From sample similarity to ensemble similarity: Probabilistic distance measures in reproducing kernel Hilbert space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28: 917-929, 2006.
13. Bernhard Schölkopf and Alexander J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2002.
14. Marco Cuturi, Kenji Fukumizu, and Jean-Philippe Vert. Semigroup kernels on measures. Journal of Machine Learning Research, 6: 11691198, 2005.
15. André F. T. Martins, Noah A. Smith, Eric P. Xing, Pedro M. Q. Aguiar, and Mário A. T. Figueiredo. Nonextensive information theoretical kernels on measures. Journal of Machine Learning Research, 10: 935-975, 2009.
16. Matthias Hein and Olivier Bousquet. Hilbertian metrics and positive definite kernels on probability measures. In AISTATS, pages 136-143, 2005.
17. Barnabás Póczos, Liang Xiong, and Jeff Schneider. Nonparametric divergence estimation with applications to machine learning on distributions. In UAI, pages 599-608, 2011.
18. Barnabás Póczos, Liang Xiong, Dougal Sutherland, and Jeff Schneider. Support distribution machines. Technical report, Carnegie Mellon University, 2012. (http://arxiv.org/abs/1202. 0302).
19. Junier B. Oliva, Willie Neiswanger, Barnabás Póczos, Jeff Schneider, and Eric Xing. Fast distribution to real regression. AISTATS (JMLR W&CP), 33: 706-714, 2014.
20. Junier Oliva, Barnabás Póczos, and Jeff Schneider. Distribution to distribution regression. ICML (JMLR W&CP), 28: 1049-1057, 2013.
21. László Györfi, Michael Kohler, Adam Krzyzak, and Harro Walk. A Distribution-Free Theory of Nonparametric Regression. Springer, New-york, 2002.
22. Samory Kpotufe. k-NN regression adapts to local intrinsic dimension. Technical report, Max Planck Institute for Intelligent Systems, 2011. (http:// arxiv.org/abs/1110.4300).
23. Thomas Görtner, Peter A. Flach, Adam Kowalczyk, and Alexander Smola. Multi-instance kernels. In ICML, pages 179-186, 2002.
24. Alain Berlinet and Christine Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, 2004.
25. Yasemin Altun and Alexander Smola. Unifying divergence minimization and statistical inference via convex duality. In COLT, pages 139-153, 2006.
26. Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13: 723-773, 2012.
27. Bharath K. Sriperumbudur, Kenji Fukumizu, and Gert R. G. Lanckriet. Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research, 12: 2389-2410, 2011.
28. Kenji Fukumizu, Francis Bach, and Michael Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. Journal of Machine Learning Research, 5: 73-99, 2004.
29. Andreas Christmann and Ingo Steinwart. Universal kernels on non-standard input spaces. In NIPS, pages 406-414, 2010.
30. Krikamol Muandet, Kenji Fukumizu, Francesco Dinuzzo, and Bernhard Schölkopf. Learning from distributions via support measure machines. In NIPS, pages 10-18, 2012.
31. Larry Wasserman. All of Nonparametric Statistics. Springer, 2006.
32. Felipe Cucker and Steve Smale. On the mathematical foundations of learning. Bulletin of the American Mathematical Society, 39: 1-49, 2002.
33. Christian Berg, Jens Peter Reus Christensen, and Paul Ressel. Harmonic Analysis on Semigroups. Springer-Verlag, 1984.
34. Bharath Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Gert Lanckriet, and Bernhard Schölkopf. Hilbert space embeddings and metrics on probability measures. Journal of Machine Learning Research, 11: 1517-1561, 2010.
35. Zhuang Wang, Liang Lan, and Slobodan Vucetic. Mixture model for multiple instance regression and applications in remote sensing. IEEE Transactions on Geoscience and Remote Sensing, 50: 2226-2237, 2012.
36. Yuchen Zhang, John C. Duchi, and Martin J. Wainwright. Divide and conquer kernel ridge regression: A distributed algorithm with minimax optimal rates. Technical report, University of California, Berkeley, 2014. (http://arxiv.org/ abs/1305.5029).
37. Zoltán Szabó, Bharath Sriperumbudur, Barnabás Póczos, and Arthur Gretton. Learning theory for distribution regression. Technical report, Gatsby Unit, University College London, 2014. (http:// arxiv.org/abs/1411.2066).
38. Michael Reed and Barry Simon. Methods of Modern Mathematical Physics-Functional Analysis. Academic Press, 1980.
39. Gabriel Nagy. Real analysis (lecture notes): Chapter III: Measure theory, Section 3: Measurable spaces and measurable maps. Technical report, Kansas State University. (http://www. math.ksu.edu/nagy/real-an/3-03-measbl. pdf).
40. K.R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, 1967.
41. John L. Kelley. General Topology. Springer, 1975.
42. John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004.
43. Zoltán Szabó. Information theoretical estimators toolbox. Journal of Machine Learning Research, 15: 283-287, 2014.