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Information Geometry of U-Boost and Bregman Divergence

noboru.murata{at}eb.waseda.ac.jp, School of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan

Takashi Takenouchi

ttakashi{at}ism.ac.jp, Department of Statistical Science, Graduate University of Advanced Studies, Minato, Tokyo 106-8569, Japan

Takafumi Kanamori

kanamori{at}is.titech.ac.jp, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan

Shinto Eguchi

eguchi{at}ism.ac.jp, Institute of Statistical Mathematics, Japan, and Department of Statistical Science, Graduate University of Advanced Studies, Minato, Tokyo 106-8569, Japan

We aim at an extension of AdaBoost to U-Boost, in the paradigm to build a stronger classification machine from a set of weak learning machines. A geometric understanding of the Bregman divergence defined by a generic convex function U leads to the U-Boost method in the framework of information geometry extended to the space of the finite measures over a label set. We propose two versions of U-Boost learning algorithms by taking account of whether the domain is restricted to the space of probability functions. In the sequential step, we observe that the two adjacent and the initial classifiers are associated with a right triangle in the scale via the Bregman divergence, called the Pythagorean relation. This leads to a mild convergence property of the U-Boost algorithm as seen in the expectation-maximization algorithm. Statistical discussions for consistency and robustness elucidate the properties of the U-Boost methods based on a stochastic assumption for training data.

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