Kernel-based non-parametric Bayesian clustering models

该建议基于这样一种信念，即复杂数据结构建模成功的关键在于在贝叶斯非参数模型中嵌入基于核或核诱导的方法。贝叶斯范式为结构化数据约束建模提供了丰富的框架，而基于核的方法则具有灵活性（例如局部、空间和/或时间适应）。提出的两个主要成分是引入局部自适应核，使用离散吉布斯随机场（如自逻辑模型和波茨模型）和确定性点过程作为分类和回归模型中未知聚类数据结构的先验或惩罚。我们可以证明自适应核与核密度和回归估计量有关。因此，为这些估计器开发的理论可以适用于更复杂的学习算法。离散吉布斯随机场提供了一个有趣的替代常用的狄利克雷过程先验。它们的集群被链接到数据图上。因此，离散图的丰富理论可以用来指导参数的选择和估计。决定性点过程提供了另一种方法来指导簇的发现。Wang-Landau算法等计算技术将以一种有效的方式扩展到我们的模型中。例如，在某些情况下，可以将多维参数相关量分解为低维参数量，以降低随机抽样算法的复杂性。贝叶斯变分技术，如平均场近似，将明确地发展，以解决我们的模型的计算需求。事实上，对于波茨模型的特殊情况，物理学文献中已经提出了其他几种更好的近似变分技术。我们将研究、适应和扩展这些技术到更复杂的模型中，用于分类和回归的聚类。

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.