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

本提案中描述的工作的主要目标是介绍 β 随机矩阵系综类并对其进行彻底的理论和数值分析。研究人员将分别对应于情况 1、2、4 的经典实数、复数和四元数随机矩阵系综推广并扩展到任何正 beta。无限随机矩阵理论中的许多结果和有限随机矩阵理论中的一些结果表明，使用 beta 作为连续参数是合理的。该提案中的关键新概念是 beta 随机变量的概念，该对象在所有实际目的中都表现为实数上的 beta 维代数。为了发展这一理论，PI 使用了“幽灵”和“影子”的概念。 “幽灵”是β维随机变量，“影子”是可以采样的派生实数或复数。除了理论结果的推导之外，该项目的一个主要目标是提供这些随机矩阵集成的计算算法。从生物信息学、基因组学（群体分类）到无线通信（网络容量优化）和军事应用（自动目标分类）的大量实际应用都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，并为这些领域带来新的应用和研究方向。

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.