CAREER: An integrated probabilistic approach to discrete and continuous extremal problems via information theory

职业：通过信息论解决离散和连续极值问题的综合概率方法

• 批准号: 1056996
• 负责人: Mokshay Madiman
• 金额: $ 60.94万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-15 至 2014-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1056996&HistoricalAwards=false
• 关键词: CAREER; integrated; probabilistic; approach; discrete

Mathematics abounds with extremal problems problems where the goal is to minimize some functional applied to a class of objects under some constraint, identify the extremal objects, and investigate the stability of extrema. Relevant examples range from some in the continuous world (isoperimetric phenomena in convex geometry, functional analytic inequalities), to some in thediscrete world (structural phenomena in additive combinatorics), and some in both (maximum entropy problems in statistics, limit theorems in probability). A natural language for all of these problem classes is probability, and, although not obvious, information theory. The project will develop new formulations of extremal problems from each of these fields in terms of information-theoretic inequalities, and then use a variety of tools from analysis, probability, convex geometry, combinatorics, and information theory, to make progress on them. The unifying nature of the perspective adopted will bridge discrete and continuous problems using a common set of tools, and enable significant cross-fertilization. Furthermore, some of the information-theoretic inequalities developed, combined with statistical decision theory, will be applied to novel statistical challenges involving multiple players that arise in engineering, economics, and biology (specifically, theoretical foundations for the problems of data pricing and distributed inference). The project will use information-theoretic thinking to make advances on challenging mathematical problems from the three seemingly disparate fields of convex geometry, arithmetic combinatorics, and probability.Apart from the intrinsic significance of these areas within mathematics, they have much practical significance - convex geometry finds applications in medical tomography, arithmetic combinatorics in computer science, and probability is ubiquitous as the foundation of statistical inference. The interpretability and unifying nature of the proposed research, and the diversity of tools it uses, create wonderful opportunities for student motivation. Newly developed courses and a resource website on information theoretic approaches to extremal problems will exploit these opportunities. The investigator will disseminate key findings through survey articles, organize an interdisciplinary workshop, and communicate the excitement of research through non-academic public lectures to attract promising students to the mathematical sciences. The applied component of the research would also have broad impact, by contributing to how data collectors and vendors come up with pricing mechanisms (e.g., for pricing of advertisements by search engines), and by improving the way networks of sensors collect and use data for various applications (e.g., for disaster recovery coordination or smart kindergartens).

数学中充满了极值问题，其目标是在某些约束下最小化应用于一类对象的某些函数，识别极值对象，并研究极值的稳定性。 相关的例子包括连续世界中的一些（凸几何中的等周现象、泛函分析不等式）、离散世界中的一些（加性组合学中的结构现象）以及两者中的一些（统计学中的最大熵问题、概率中的极限定理）。 所有这些问题类别的自然语言是概率，以及信息论（尽管不明显）。 该项目将从信息论不等式的角度开发每个领域的极值问题的新表述，然后使用分析、概率、凸几何、组合学和信息论等多种工具，在这些问题上取得进展。 所采用的视角的统一性将使用一组通用工具来弥合离散和连续的问题，并实现显着的交叉融合。 此外，一些信息论不等式与统计决策理论相结合，将应用于工程、经济学和生物学领域出现的涉及多个参与者的新统计挑战（特别是数据定价和分布式推理问题的理论基础）。 该项目将利用信息论思维，从凸几何、算术组合学和概率这三个看似不同的领域，在具有挑战性的数学问题上取得进展。除了这些领域在数学中的内在意义外，它们还具有很大的实际意义——凸几何在医学断层扫描、计算机科学中的算术组合学中得到应用，而概率作为数学的基础无处不在。 统计推断。 拟议研究的可解释性和统一性，以及它使用的工具的多样性，为学生的积极性创造了绝佳的机会。 新开发的关于极端问题信息论方法的课程和资源网站将利用这些机会。 研究者将通过调查文章传播重要发现，组织跨学科研讨会，并通过非学术公开讲座传达研究的兴奋点，以吸引有前途的学生进入数学科学领域。 研究的应用部分也将产生广泛的影响，通过促进数据收集者和供应商如何提出定价机制（例如，通过搜索引擎对广告定价），以及改进传感器网络收集和使用各种应用数据的方式（例如，用于灾难恢复协调或智能幼儿园）。