CAREER: Geometric Algorithms For Data Analysis In Spaces Of Distributions

在分析大型数据集时，自然会出现分布集合。由于存储这些数据的一小部分之外的所有数据是不切实际的，因此通常使用分布表示以紧凑的形式总结数据。例如，语料库中的文档通常由关键词出现频率的归一化向量表示，图像由梯度特征上的直方图表示，语音信号由频域上的谱密度表示。将数据集表示为分布的集合，可以通过统计学、学习理论和信息论的强大概念进行分析。信念强度、信息内容和模式似然等概念用于从数据中提取意义和结构，并使用Kullback-Leibler距离及其父类Bregman散度等信息度量进行量化。通过将信息丢失和传递的抽象概念与具体的几何概念（如距离）联系起来，这些度量方法以传统度量方法无法做到的方式捕捉数据中的意义。然而，它们缺乏对称性和三角不等式等特性，而这些特性是应用传统几何算法进行数据分析的基本要求。在这个项目中，PI将开发一个系统的、严格的和全球性的算法框架来操纵这些距离。这个框架将为有效和准确地分析分布空间的数据提供基础，并将在广泛的应用程序中对分析问题产生更深入的见解。

Collections of distributions arise naturally when analyzing large data sets. Since it is impractical to store all but a small fraction of such data, distributional representations are typically used to summarize the data in compact form. For example, a document in a corpus is typically represented by a normalized vector of frequencies of occurrence of keywords, an image is represented by a histogram over gradient features and speech signals are represented by spectral densities over a frequency domain.Representing data sets as collections of distributions enables analysis via powerful concepts from statistics, learning theory and information theory. Concepts like strength of belief, information content, and pattern likelihood are used to extract meaning and structure from the data and are quantified using information measures like the Kullback-Leibler distance and its parent class, the Bregman divergences.These measures capture meaning in data in a manner that traditional metrics cannot, by connecting abstract notions of information loss and transfer with concrete geometric notions like distances. However, they lack properties like symmetry and the triangle inequality that are essential requirements for the application of traditional geometric algorithms for data analysis.In this project, the PI will develop a systematic, rigorous and global algorithmic framework for manipulating these distances. This framework will provide the foundation for efficient and accurate data analysis of spaces of distributions, and will lead to deeper insights into analysis problems across a wide range of applications.