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Computing gaussian mixture models with EM using equivalence constraints.pdf

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Computing gaussian mixture models with EM using equivalence constraints

Computing Gaussian Mixture Models with EM using Side-Information Noam Shental fenoam@cs.huji.ac.il Aharon Bar-Hillel aharonbh@cs.huji.ac.il Tomer Hertz tomboy@cs.huji.ac.il Daphna Weinshall daphna@cs.huji.ac.il School of Computer Science and Engineering and the Center for Neural Computation, The Hebrew University of Jerusalem, 91904 Jerusalem, ISRAEL Abstract Estimation of Gaussian mixture models is an efficient and popular technique for clustering and density estimation. An EM procedure is widely used to estimate the model parame- ters. In this paper we show how side informa- tion in the form of equivalence constraints can be incorporated into this procedure, leading to improved clustering results. Equivalence constraints are prior knowledge concerning pairs of data points, indicating if the points arise from the same source (positive con- straint) or from different sources (negative constraint). Such constraints can be gath- ered automatically in some learning prob- lems, and are a natural form of supervision in others. We present a closed form EM procedure for handling positive constraints, and a Generalized EM procedure using a Markov net for the incorporation of negative constraints. Using publicly available data sets we demonstrate that such side informa- tion may lead to considerable improvement in clustering tasks, and that our algorithm is preferable to another suggested method using this type of side information. Keywords: Learning from partial knowledge, semi- supervised learning, Gaussian mixture models, clus- tering. 1. Introduction We are used to thinking about learning from labels as supervised learning, and learning without labels as un- supervised learning, where ’supervised’ implies a need for human intervention. However, in unsupervised learning we are not limited to using data statistics only. Similarly supervised learning is not limited to using labels. In this work we focus on semi-supervised learning using side-information, which is not given