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Nonextensive Entropic Kernels



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Nonextensive Entropic Kernels Andre F T Martins afm cs cmu edu Ma rio A T Figueiredo mario figueiredo lx it pt Pedro M Q Aguiar aguiar isr ist utl pt Noah A Smith nasmith cs cmu edu Eric P Xing epxing cs cmu edu School of Computer Science Carnegie Mellon University Pittsburgh PA USA Instituto de Telecomunicac o es Instituto de Sistemas e Robo tica Instituto Superior Te cnico Lisboa Portugal Abstract Positive definite kernels on probability measures have been recently applied in structured data classification problems Some of these kernels are related to classic information theoretic quantities such as mutual information and the Jensen Shannon divergence Meanwhile driven by recent advances in Tsallis statistics nonextensive generalizations of Shannon s information theory have been proposed This paper bridges these two trends We introduce the Jensen Tsallis q difference a generalization of the Jensen Shannon divergence We then define a new family of nonextensive mutual information kernels which allow weights to be assigned to their arguments and which includes the Boolean Jensen Shannon and linear kernels as particular cases We illustrate the performance of these kernels on text categorization tasks 1 Introduction There has been recent interest in kernels on probability distributions to tackle several classification problems Moreno et al 2003 Jebara et al 2004 Hein Bousquet 2005 Lafferty Lebanon 2005 Cuturi et al 2005 By mapping data points to fitted distributions in a parametric family where a kernel is defined a kernel is automatically induced on the original input space In text categorization this appears as an alternative to the Euclidean geometry inherent to Appearing in Proceedings of the 25 th International Conference on Machine Learning Helsinki Finland 2008 Copyright 2008 by the author s owner s the usual bag of words vector representations In fact approaches that map data to a statistical manifold where well motivated non Euclidean metrics may be defined



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