Statistical learning via multivariate density estimation

该项目的总体目标是开发多维度密度估计方法，并根据这一方法开发新工具，以解决数据压缩、图像分析和图形模型推理方面的选定问题。密度估计是统计学中的一个基本问题，但传统的方法，如核密度估计不适合处理大型多元数据集在当前的应用。本项目的研究主要集中在多元密度估计的方法和应用上。通过为这个问题创建有效的方法，该项目还将有利于应用统计和机器学习中的许多其他研究问题，其中密度估计可以用作解决方案的构建块，例如图像分割，数据压缩和网络建模。 具体来说，该项目将解决如何推断样本空间的分区的问题，这将揭示底层数据分布的结构。将基于贝叶斯非参数方法从观察数据中学习分区，该方法对待估计的分布施加最小假设。这种推理的有效和可扩展的算法将被设计用于多维度的大型数据集的分析。估计的理论性质，如渐近一致性和收敛速度，也将被调查。

The overall goal of the project is to develop methodologies of density estimation in multiple dimensions, and to develop new tools based on this methodology for selected problems in data compression, image analysis and graphical model inference. Density estimation is a fundamental problem in statistics but traditional approaches such as kernel density estimation are not well suited to handle the large multivariate data sets in current applications. The research in this project is centered on the methodology and application of multivariate density estimation. By creating effective methods for this problem, this project will also benefit many other research problems in applied statistics and machine learning where density estimation can be used as a building block for the solution, for example, image segmentation, data compression and network modeling. Specifically, the project will address the question of how to infer a partition of the sample space that will reveal the structure of the underlying data distribution. The partition will be learned from the observed data based on a Bayesian nonparametric approach which imposes minimal assumptions on the distribution to be estimated. Efficient and scalable algorithms for such inferences will be designed for the analysis of large data sets in multiple dimensions. The theoretical properties of the estimates, such as asymptotic consistency and convergence rates, will also be investigated.