Hilbert Space Methods for Statistics and Probability

15 May 2019 13:45

For representing possible regression functions (very natural); for representing possible probability distributions in density estimation, independence tests, and two-sample tests (less natural but important).

Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Bernhard Schölkopf, Gert R.G. Lanckriet, "Hilbert Space Embeddings and Metrics on Probability Measures", Journal of Machine Learning Research 11 (2010): 1517--1561
Grace Wahba, Spline Models for Observational Data

Eduardo Corona, Terran Lane, Curtis Storlie, Joshua Neil, "Using Laplacian Methods, RKHS Smoothing Splines and Bayesian Estimation as a framework for Regression on Graph and Graph Related Domains" [Technical report, University of New Mexico Computer Science, 2008-06, PDF]
Kenji Fukumizu, Le Song, Arthur Gretton, "Kernel Bayes' rule", arxiv:1009.5736

Nancy Heckman, "The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy", arxiv:1111.1915
Alexander Jung, Sebastian Schmutzhard, Franz Hlawatsch, "The RKHS Approach to Minimum Variance Estimation Revisited: Variance Bounds, Sufficient Statistics, and Exponential Families", arxiv:1210.6516
Vern I. Paulsen and Mrinal Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces `
James Robins and Aad van der Vaart, "Adaptive nonparametric confidence sets", Annals of Statistics 34 (2006): 229--253, arxiv:math/0605473 ["We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models."]
Christopher G. Small and D. L. McLeish, Hilbert Space Methods in Probability and Statistical Inference
Bharath K. Sriperumbudur, Kenji Fukumizu, Gert R. G. Lanckriet, "Universality, Characteristic Kernels and RKHS Embedding of Measures", arxiv:1003.0887