Probability Link Models with Symmetric Information Divergence

This paper introduces link functions for transforming one probability distribution to another such that the Kullback-Leibler and R\'enyi divergences between the two distributions are symmetric. Two general classes of link models are proposed. The first model links two survival functions and is applicable to models such as the proportional odds and change point, which are used in survival analysis and reliability modeling. A prototype application involving the proportional odds model demonstrates advantages of symmetric divergence measures over asymmetric measures for assessing the efficacy of features and for model averaging purposes. The advantages include providing unique ranks for models and unique information weights for model averaging with one-half as much computation requirement of asymmetric divergences. The second model links two cumulative probability distribution functions. This model produces a generalized location model which are continuous counterparts of the binary probability models such as probit and logit models. Examples include the generalized probit and logit models which have appeared in the survival analysis literature, and a generalized Laplace model and a generalized Student-$t$ model, which are survival time models corresponding to the respective binary probability models. Lastly, extensions to symmetric divergence between survival functions and conditions for copula dependence information are presented.

Non-negative matrix factorization based on generalized dual divergence

A theoretical framework for non-negative matrix factorization based on generalized dual Kullback-Leibler divergence, which includes members of the exponential family of models, is proposed. A family of algorithms is developed using this framework and its convergence proven using the Expectation-Maximization algorithm. The proposed approach generalizes some existing methods for different noise structures and contrasts with the recently proposed quasi-likelihood approach, thus providing a useful alternative for non-negative matrix factorizations. A measure to evaluate the goodness-of-fit of the resulting factorization is described. This framework can be adapted to include penalty, kernel and discriminant functions as well as tensors.