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Hilbert Space Embedding of Probability Measures: Theory and Applications

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Modern scientific fields (e.g., social sciences, bioinformatics, biomedical imaging and genomics) routinely deal with high-dimensional and highly complex information, including data from non-Euclidean spaces such as trees, graphs and strings. In this talk, I will introduce a functional analytic method for representing and analyzing high-dimensional and complex data by embedding probability measures into a reproducing kernel Hilbert space (RKHS). Such embeddings can be seen as a generalization of characteristic function associated with a probability measure. This generalization allows us to represent and compare random variables on domains more general than R^n (including graphs, strings, and groups), which can then be exploited in many statistical applications like homogeneity/independence/goodness-of-fit testing, feature selection and density estimation. I will discuss various theoretical questions related to distribution embeddings, for example, when is the embedding injective and how it is related to the properties of the RKHS ? I will then present two applications of the embedding in two-sample testing and mixture sieve density estimation. Finally, I will conclude by discussing some open problems and ongoing/future research directions.

Joint work with Kenji Fukumizu (The Institute of Statistical Mathematics), Arthur Gretton (Gatsby Computational Neuroscience Unit, UCL ), Gert Lanckriet (UC San Diego) and Bernhard Scholkopf (Max Planck Institute for Intelligent Systems)

This talk is part of the Statistics series.

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