Clustering with Bregman Divergences
The Journal of Machine Learning Research
Mixed Bregman Clustering with Approximation Guarantees
ECML PKDD '08 Proceedings of the European conference on Machine Learning and Knowledge Discovery in Databases - Part II
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Bregman divergences in the (m×k)-partitioning problem
Computational Statistics & Data Analysis
Adaptive fuzzy filtering in a deterministic setting
IEEE Transactions on Fuzzy Systems
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IEEE Transactions on Information Theory
Sided and symmetrized Bregman centroids
IEEE Transactions on Information Theory
Aggregation functions based on penalties
Fuzzy Sets and Systems
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Journal of Multivariate Analysis
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Pattern Recognition Letters
Information, Divergence and Risk for Binary Experiments
The Journal of Machine Learning Research
Bregman clustering for separable instances
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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We consider the problem of predicting a random variable X from observations, denoted by a random variable Z. It is well known that the conditional expectation E[X|Z] is the optimal L2 predictor (also known as "the least-mean-square error" predictor) of X, among all (Borel measurable) functions of Z. In this orrespondence, we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor. We show that E[X|Z] is the optimal predictor for all Bregman loss functions (BLFs), of which the L2 loss function is a special case. Moreover, under mild conditions, we show that the BLFs are exhaustive, i.e., if for every random variable X, the infimum of E[F(X,y)] over all constants y is attained by the expectation E[X], then F is a BLF.