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

再现核希尔伯特空间嵌入的测量：统计学习的理论和应用

基本信息

批准号：
1713011
负责人：
Bharath Sriperumbudur
金额：
$ 17.83万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1713011&HistoricalAwards=false
关键词：
Reproducing Kernel Hilbert Space Embedding

项目摘要

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.

在这个项目中考虑的统计估计和推断问题出现在依赖于统计研究的许多科学和工程领域。其中一些具有高度社会影响的领域包括社会和行为研究，法医学，早期发现秘密通信和数据安全漏洞，用于识别传染病爆发的预警系统，儿童认知发展研究和药物发现等。在许多这些领域，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，即使这些假设是有道理的，它们也需要得到检验。该研究项目调查了这些现有方法的强大替代方案，旨在发展对相同方法的基本理论理解，从而导致新的统计应用。核方法是一类统计方法学，由于其能够处理高维和非欧几里德数据，在统计学习中得到了广泛的应用。该方法的核心思想是将观测数据映射到一个函数空间，称为再生核希尔伯特空间（RKHS），它允许捕获数据中的非线性关系。该项目涉及通过在RKHS中嵌入概率测度来推广该方法的理论和方法研究。这种概括在统计学习问题中具有广泛的适用性，例如非参数假设检验，密度估计和分布回归，这些将在本项目中进行探索。在理论方面，将考虑核嵌入的内射性的表征。虽然这样的特征是很好地理解的核定义在局部紧阿贝尔群和紧非阿贝尔群，这个项目将调查的内射嵌入的非标准空间，如核空间，空间的图形，和正定锥。已知嵌入的内射性与RKHS在逼近某类函数时的丰富性有关。该研究将研究这种近似的速度，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时至关重要。内射嵌入引入了一个度量，称为概率空间上的核距离，它被定义为两个概率测度的核嵌入之间的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（缺点）优势。这些理论研究有一个应用对应，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关键作用。对于这些问题，研究人员计划开发计算效率的估计与理论保证。总体而言，该项目旨在开发一个全面的RKHS嵌入概率测度理论，并将其应用于统计学习问题。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

DOI：
10.1007/s10208-018-09407-7
发表时间：
2017-09
期刊：
Foundations of Computational Mathematics
影响因子：
3
作者：
Motonobu Kanagawa;Bharath K. Sriperumbudur;K. Fukumizu
通讯作者：
Motonobu Kanagawa;Bharath K. Sriperumbudur;K. Fukumizu

On kernel derivative approximation with random Fourier features.

关于具有随机傅立叶特征的核导数近似。

DOI：
发表时间：
2019
期刊：
The 22nd International Conference on Artificial Intelligence and Statistics
影响因子：
0
作者：
Szabo, Zoltan;Sriperumbudur, Bharath K.
通讯作者：
Sriperumbudur, Bharath K.

Characteristic and Universal Tensor Product Kernels

DOI：
10.13140/rg.2.2.27112.37120
发表时间：
2017-08
期刊：
J. Mach. Learn. Res.
影响因子：
0
作者：
Z. Szabó;Bharath K. Sriperumbudur
通讯作者：
Z. Szabó;Bharath K. Sriperumbudur

Gain with no Pain: Efficiency of Kernel-PCA by Nyström Sampling

轻松获得收益：通过 Nyström 采样提高内核 PCA 的效率

DOI：
发表时间：
2020
期刊：
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics
影响因子：
0
作者：
Sterge, N;Sriperumbudur, B. K.;Rosasco, L.;and Rudi, A.
通讯作者：
and Rudi, A.