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An overview of covariance operators in Hilbert space, and their applications

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If you have a question about this talk, please contact Karsten Borgwardt.

Many problems in unsupervised learning require the analysis of features of probability distributions. In this talk, we deal with the problem of measuring dependence between random variables, using the covariance between maps of these variables to spaces of features. When the feature spaces are universal reproducing kernel Hilbert spaces (RKHSs), it can be shown that the covariance is zero only when the variables are independent.

A different perspective on these operators arises when one seeks to manipulate variables so as to maximise their dependence (as measured by feature space covariance), rather than minimising it. In the case of feature selection, we would choose those features that maximise the dependence with respect to particular target variables. Several well-known feature selection algorithms can be recovered through an appropriate feature space choice. Finally, covariance operators may be combined to give measures of conditional covariance, which may be used to measure conditional dependence.

This talk is part of the Machine Learning @ CUED series.

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