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

这项工作的主要目标是介绍和执行一类贝塔随机矩阵系综的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推广到任何正的贝塔。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β作为连续参数是合理的。这一提议中的关键新概念是贝塔随机变量的概念，这是一个在所有实际目的中表现为实数上的贝塔维代数的对象。为了发展这一理论，PI们使用了“幽灵”和“阴影”的概念。“幽灵”是贝塔维度的随机变量，而“影子”是可以采样的派生的实数或复数。除了理论结果的推导，这个项目的一个主要目标是提供计算这些随机矩阵集成的算法。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)，大量的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，并使这些领域的新应用和研究方向成为可能。

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.