Collaborative Research: Theory and Algorithms for Beta Random Matrices: The Random Matrix Method of "Ghosts" and "Shadows"

合作研究：β随机矩阵的理论与算法：“鬼”与“影”的随机矩阵方法

• 批准号: 1016125
• 负责人: Alan Edelman
• 金额: $ 28.01万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016125&HistoricalAwards=false
• 关键词: Collaborative; Research; Theory; Algorithms; Beta

The main goal of the work described in this proposal is to introduce and perform a thorough theoretical and numerical analysis of the class of beta random matrix ensembles. The investigators will generalize and extend to any postive beta the classical real, complex, and quaternion random matrix ensembles which correspond to the cases 1, 2, 4, respectively. Already many results in infinite random matrix theory and a few results in finite random matrix theory suggest that the use of beta as a continuous parameter is reasonable. The key new concept in this proposal is the notion of a beta-random variable, an object which, for all practical purposes behaves as a beta-dimensional algebra over the reals. To develop the theory the PIs use the notions of "ghosts" and "shadows". A "ghost" is a beta-dimensional random variable and a "shadow" is a derived real or complex quantity that can be sampled. Along with the derivation of theoretical results, a major goal of this project is to provide algorithms for computation with these random matrix ensembles.A vast number of practical application ranging from bioinformatics, and genomics (population classification) to wireless communications (network capacity optimization) and military applications (automatic target classification) rely on the methods of multivariate statistics and in turn on random matrix theory. The proposed research will provide new algorithmic and theoretical tools for these applications as well as enable new applications and research directions in these fields.

在这个建议中所描述的工作的主要目标是引入和执行一个彻底的理论和数值分析的类β随机矩阵合奏。研究人员将推广和扩展到任何正β的经典真实的，复杂的，和四元数随机矩阵合奏，分别对应的情况1，2，4。在无限随机矩阵理论和有限随机矩阵理论中已经有许多结果表明使用β作为连续参数是合理的。这个提议中的关键新概念是β-随机变量的概念，一个对象，对于所有实际目的来说，它表现为实数上的β维代数。为了发展这个理论，PI使用了“幽灵”和“阴影”的概念。“鬼”是β维随机变量，而“影”是可以采样的导出的真实的或复量。沿着理论结果的推导，本项目的一个主要目标是提供这些随机矩阵集合的计算算法。从生物信息学、基因组学（种群分类）到无线通信（网络容量优化）和军事应用（自动目标分类）的大量实际应用依赖于多元统计方法，进而依赖于随机矩阵理论。拟议的研究将为这些应用提供新的算法和理论工具，并在这些领域实现新的应用和研究方向。