CAREER: Foundations of Information Theory: Information Inequalities and Dimension-Free Phenomena

源于熵和互信息的信息不平等构成了香农关于通信、数据传输和存储的数学理论的基础，其巨大的影响开创了现代信息时代。虽然最初理论上限定了数据可能在不完美的信道上传输的速率，或者信息源可能被压缩的距离，但这些不平等的基本性质导致了它们在从计算机科学到物理学等定量领域的广泛应用。这可能部分归因于这些不平等是无维的，也就是说，估计的清晰度不会随着数据维度的降低而降低，从而使它们适合在高维环境中进行分析和推理，这是统计、优化和数据科学中现代问题的特征。从广义上讲，该项目旨在进一步阐明信息理论的这些基础基础，并将其扩展到现代统计和数据分析问题中，这些问题寻求从大型数据集中揭示信息。该研究与一个在多个层面整合研究和教育的计划相结合：该项目将在统计学、信息论和数学的界面上培训研究人员，为他们进入数据科学的学术和工业生涯做好准备，并随着国家优先事项的变化熟练地适应新领域。其他目标是通过举办专题讲习班和教程促进更广泛的研究界内的合作。这个项目将对信息不平等进行系统的调查。这一目标将通过一项综合研究议程来实现，该议程力求：(i)量化高维统计现象；（ii）描述信息不平等的极值性质；（iii）通过将图和其他离散空间上基于信息和运输的量联系起来，发现新的集中和等周现象。通过这些目标，该项目将促进信息理论的影响，跨越传统的界限，并丰富所涉及的知识问题。考虑到量化统计和概率现象在整个科学和工程中的内在重要性，研究结果将产生广泛而持久的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Information inequalities stemming from entropy and mutual information form the basis for Shannon's mathematical theory of communication, data transmission and storage, the vast consequences of which have ushered in the modern information age. While originally bounding theoretically the rate at which data may be transmitted over an imperfect channel, or how far an information source may be compressed, the foundational nature of these inequalities has led to their widespread applications in quantitative fields ranging from computer science to physics. This may be attributed, in part, to these inequalities being dimension-free, that is, the sharpness of the estimate does not degrade with the data dimension, thus making them suitable for analysis and inference in high-dimensional settings that are characteristic of modern problems in statistics, optimization and data science. Broadly speaking, this project seeks to further elucidate these foundational underpinnings of information theory, and to extend their applicability to modern problems in statistics and data analysis that seek to uncover information from large data sets. The research is coupled with a plan to integrate research and education at multiple levels: the project will train researchers at the interface of statistics, information theory and mathematics, preparing them to enter academic and industrial careers in data science, and skillfully adapt to new fields as national priorities change. Other aims are to promote collaboration within the broader research community through development of thematic workshops and tutorials. This project will undertake a systematic investigation of information inequalities. This goal will be achieved through an integrated research agenda that seeks to: (i) quantify high-dimensional statistical phenomena; (ii) characterize extremal properties of information inequalities; and (iii) discover new concentration and isoperimetric phenomena through linking information- and transportation-based quantities on graphs and other discrete spaces. Through these aims, this project will promote the influence of information theory across traditional boundaries and enrich the set of intellectual questions addressed. Given the intrinsic importance of quantifying statistical and probabilistic phenomena throughout science and engineering, the results will have broad and lasting impact.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.