CIF: Small: Collaborative Research: Inference of Information Measures on Large Alphabets: Fundamental Limits, Fast Algorithims, and Applications

信息论的一个关键任务是描述压缩、通信和涉及信息存储、传输和处理的更一般的操作问题的基本性能限制。这种表征通常是在信息测度方面，其中最基本的是香农熵和互信息。除了在传统的信息论领域中扮演着重要的操作角色外，信息度量在许多统计建模和机器学习任务中也有许多应用。各种现代数据分析应用程序处理的数据集自然被视为来自大域中概率分布的样本。由于典型的大字母表大小和资源限制，从业者在从语料库语言学到神经科学的应用中面临着欠采样的困难。该项目的主要目标之一是发展基于一套新的数学工具的一般理论，这将有助于构建和分析大字母信息度量的最佳估计。该项目的另一个主要方面是将新的理论方法整合到机器学习算法中，从而显著影响当前现实世界的学习实践。这个项目的成功完成将会产生使能技术和实用方案——应用范围从分析神经反应数据到学习图形模型——比现有的技术更接近基本性能极限。该项目的研究成果将丰富现有的大数据分析课程。一个新的课程致力于高维统计推断，解决估计大字母数据的深度将创建和提供。关于该项目的主题和研究结果的研讨会将在斯坦福大学和UIUC大学组织和举办。一个全面的近似理论方法来估计分布函数的大字母表将开发通过计算效率的程序基于最佳多项式近似，具有可证明的基本最优性保证。基于高维统计文献，我们的关键观察是，虽然估计分布本身需要样本量与字母大小线性缩放，但有可能准确估计分布的函数，如熵或互信息，具有亚线性样本复杂性。这需要通过开发比最大似然（插件）估计更复杂的方法来超越传统智慧。该项目的另一个主要方面是将新的理论方法转化为高度可扩展和高效的机器学习算法，从而显着影响当前现实世界的学习实践，并显着提高几种最流行的机器学习应用程序的性能，例如依赖互信息估计的学习图形模型。

A key task in information theory is to characterize fundamental performance limits in compression, communication, and more general operational problems involving the storage, transmission and processing of information. Such characterizations are usually in terms of information measures, among the most fundamental of which are the Shannon entropy and the mutual information. In addition to their prominent operational roles in the traditional realms of information theory, information measures have found numerous applications in many statistical modeling and machine learning tasks. Various modern data-analytic applications deal with data sets naturally viewed as samples from a probability distribution over a large domain. Due to the typically large alphabet size and resource constraints, the practitioner contends with the difficulty of undersampling in applications ranging from corpus linguistics to neuroscience. One of the main goals of this project is the development of a general theory based on a new set of mathematical tools that will facilitate the construction and analysis of optimal estimation of information measures on large alphabets. The other major facet of this project is the incorporation of the new theoretical methodologies into machine learning algorithms, thereby significantly impacting current real-world learning practices. Successful completion of this project will result in enabling technologies and practical schemes - in applications ranging from analysis of neural response data to learning graphical models - that are provably much closer to attaining the fundamental performance limits than existing ones. The findings of this project will enrich existing big data-analytic curricula. A new course dedicated to high-dimensional statistical inference that addresses estimation for large-alphabet data in depth will be created and offered. Workshops on the themes and findings of this project will be organized and held at Stanford and UIUC. A comprehensive approximation-theoretic approach to estimating functionals of distributions on large alphabets will be developed via computationally efficient procedures based on best polynomial approximation, with provable essential optimality guarantees. Rooted in the high-dimensional statistics literature, our key observation is that while estimating the distribution itself requires the sample size to scale linearly with the alphabet size, it is possible to accurately estimate functionals of the distribution, such as entropy or mutual information, with sub-linear sample complexity. This requires going beyond the conventional wisdom by developing more sophisticated approaches than maximal likelihood (?plug-in?) estimation. The other major facet of this project is translating the new theoretical methodologies into highly scalable and efficient machine learning algorithms, thereby significantly impacting current real-world learning practices and significantly boosting the performance in several of the most prevalent machine learning applications, such as learning graphical models, that rely on mutual information estimation.