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

信息不平等源于熵和互信息形成香农的通信，数据传输和存储的数学理论的基础，其巨大的后果已经迎来了现代信息时代。虽然最初在理论上限制了数据在不完美的信道上传输的速率，或者信息源可以被压缩多远，但这些不等式的基本性质导致了它们在从计算机科学到物理学的定量领域的广泛应用。这可能部分归因于这些不等式是无量纲的，即估计的锐度不会随着数据维度而降低，从而使它们适合于在高维环境中进行分析和推断，这是统计学，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本基础，并将其适用性扩展到统计和数据分析中的现代问题，这些问题旨在从大型数据集中揭示信息。该研究与多层次整合研究和教育的计划相结合：该项目将培训统计学，信息论和数学接口的研究人员，使他们能够进入数据科学的学术和工业职业生涯，并随着国家优先事项的变化熟练地适应新领域。其他目的是通过举办专题讲习班和辅导班，促进更广泛的研究界之间的合作。该项目将对信息不平等进行系统调查。这一目标将通过一个综合研究议程来实现，该议程力求：㈠量化高维统计现象; ㈡描述信息不平等的极端性质; ㈢通过将图形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，这个项目将促进信息理论的影响力跨越传统的界限，并丰富了一套知识问题的解决。 鉴于量化统计和概率现象在整个科学和工程中的内在重要性，其结果将产生广泛而持久的影响。该奖项反映了NSF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.