Kernel-based non-parametric Bayesian clustering models

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This proposal is based on the belief that the key to the success of modeling complex data structures lies in the embedding of kernel-based or kernel-induced methods within Bayesian non-parametric models. The Bayesian paradigm gives a rich framework for modeling structural data constraints, whereas kernel-based methods incorporates flexibility (e.g. local, spatial and/or temporal adaptation). Two main ingredients proposed are the introduction of locally adaptive kernels, and the use of discrete Gibbs random fields (such as the auto-logistic and Potts models) and determinantal point processes as priors or penalization for the unknown clustering data structure in classification and regression models. We can show that the adaptive kernels are linked to kernel density and regression estimators. Consequently, the theory developed for these estimators can be adapted to more complex learning algorithms. Discrete Gibbs random fields offer an interesting alternative to the commonly used Dirichlet process priors. Their clusters are linked to data graphs. Therefore, the rich theory developed for discrete graphs may be used to guide the choice of parameters and their estimation. Determinantal point processes offer another alternative to guide the discovery of clusters. Computational techniques such as the Wang-Landau algorithm will be extended to our models in an efficient way. For example, in some cases, multi-dimensional parameter dependent quantities may be factorized in lower-dimensional parameter quantities so as to reduced the complexity of the stochastic sampling algorithms. Bayesian variational techniques, such as the mean-field approximation, will be developed explicitly to address the computational needs of our models. Indeed, for the particular case of the Potts model, several other better approximating variational techniques have been suggested in the Physics literature. We will study, adapt and extend these techniques to more complex models for clustering for classification and regression.