Reproducing Kernel Hilbert Space Embedding of Measures: Theory and Applications to Statistical Learning

这个项目中考虑的统计估计和推理问题出现在许多依赖统计研究的科学和工程领域。其中一些具有高度社会影响的领域包括社会和行为研究、法医科学、秘密通信和数据安全漏洞的早期发现、用于识别传染病爆发的早期预警系统、儿童认知发展研究和药物发现等。在这些领域中的许多领域，常用的统计方法对数据生成分布做出了强有力的假设。这种简单化的方法可能是不合理的；此外，即使这些假设是有意义的，它们也需要进行测试。这项研究项目调查了这些现有方法的一个强有力的替代方法，目的是发展对这些方法的基本理论理解，导致新的统计应用。核方法是一种统计方法，由于其处理高维和非欧几里德数据的能力而在统计学习中广受欢迎。该方法的核心思想是将观测数据映射到一个称为再生核Hilbert空间(RKHS)的函数空间，该空间允许捕获数据中的非线性关系。该项目涉及通过在RKHS中嵌入概率度量来推广该方法的理论和方法研究。这一推广在统计学习问题中具有广泛的适用性，如非参数假设检验、密度估计和分布回归，这些都将在本项目中探索。在理论方面，将考虑核嵌入的内射性的刻画。虽然这样的刻画对于定义在局部紧Abel群和紧非Abel群上的核是很好理解的，但这个项目将研究非标准空间(如核空间、图空间和正定锥)的核嵌入的内射性。已知嵌入的内射性与RKHS在逼近某一类函数时的丰富性有关。这项研究将考察这种近似的速度，这在分析基于核的回归和密度估计的收敛速度以及假设检验中的分离率方面是至关重要的。内射嵌入在概率空间上引入一个度量，称为核距离，定义为两个概率度量的核嵌入之间的RKHS距离。研究人员将研究核距离与其他概率度量的关系，例如能量距离、距离协方差、f-散度和积分概率度量，以了解与这些距离相关的统计/计算(Dis)优势。这些理论研究具有应用价值，其中RKHS嵌入在无穷维指数族的概率测度回归和密度估计问题中起着至关重要的作用。对于这些问题，研究人员计划开发计算效率高的估计器，并提供理论保证。总体而言，该项目旨在发展一种全面的RKHS嵌入概率度量的理论，并将其应用于统计学习中的问题。

The statistical estimation and inference questions considered in this project appear in many areas of science and engineering that rely on statistical research. Some of these areas of high societal impact include social and behavioral research, forensic sciences, early detection of covert communications and breaches in data security, early-warning systems for identifying outbreaks of infectious diseases, cognitive development studies in children, and drug discovery, among many others. In many of these areas, commonly-used statistical methods make strong assumptions about the data-generating distribution. Such simplistic approaches may be unjustified; moreover, even if these assumptions make sense, they need to be tested. This research project investigates a powerful alternative to these existing methods and aims to develop a fundamental theoretical understanding of the same, leading to novel statistical applications. Code for algorithms that result from this project will be made publicly available for ready use.The kernel method is a class of statistical methodology that has gained popularity in statistical learning due to its ability to handle both high-dimensional and non-Euclidean data. The core idea of the method is to map observed data to a function space, called the reproducing kernel Hilbert space (RKHS), which allows capture of non-linear relationships in the data. This project concerns theoretical and methodological research on a generalization of this method by embedding probability measures in an RKHS. This generalization has wide applicability in statistical learning problems such as nonparametric hypothesis testing, density estimation, and regression on distributions, which will be explored in this project. On the theoretical front, the characterization of injectivity of kernel embedding will be considered. While such a characterization is well understood for kernels defined on locally compact Abelian groups and compact non-Abelian groups, this project will investigate the injectivity of the kernel embedding for non-standard spaces such as nuclear spaces, the space of graphs, and the positive definite cone. The injectivity of the embedding is known to be related to the richness of the RKHS in approximating a certain class of functions. The research will investigate the rate of this approximation, which turns out to be critical in analyzing the convergence rates of kernel-based regression and density estimators and separation rates in hypothesis testing. An injective embedding induces a metric, called the kernel distance on the space of probabilities, which is defined as the RKHS distance between the kernel embeddings of two probability measures. The investigator will study the relation of kernel distance to other probability metrics such as the energy distance, distance covariance, f-divergence, and integral probability metrics in order to understand the statistical/computational (dis)advantages associated with these distances. These theoretical studies have an applied counterpart, wherein the RKHS embedding plays a critical role in the problems of regression on probability measures and density estimation in infinite dimensional exponential families. For these problems, the investigator plans to develop computationally efficient estimators with theoretical guarantees. Overall, the project aims to develop a comprehensive theory of RKHS embedding of probability measures with applications to problems in statistical learning.