MSPA-MCS: New Tools for Algebro-Geometric Data Analysis

摘要：迈克尔·柯比提议：0434351该提案开发了新的代数几何工具，用于数据分析，特别是用于调查和理解计算机科学中出现的大量数据集，其中环境维度和样本数量可能很大。 该团队感兴趣的特别例子包括脑机接口（BCI）问题和计算机视觉和识别问题。 具体而言，该小组正在将割线品种（任意顺序）与确定数据集的最佳投影相关联;他们询问品种的顺序是适当的，并正在调查不同顺序品种之间的关系。 割线簇也在理解高阶张量的标准形和分解中发挥作用，这反过来又是信号处理应用（如BCI问题）的基本兴趣。 代数几何为高阶张量的算法开发提供了一个框架。 最后，关于格拉斯曼的舒伯特演算的最新发展极大地增加了这一理论的潜在适用性。 研究人员正在研究如何利用这些想法来开发有效的算法，以找到受约束的接近最佳的投影仪。 该团队正在进行的研究对数学和计算机科学界产生了独特的影响。 通过将来自看似不同专业领域的研究人员聚集在一起，它旨在这些领域之间的交叉施肥。 计算机科学提供了该团队正在开发新算法的问题。 相反，该团队希望从这项研究所解决的实际问题中产生新的数学模型。 此外，研究生（都在攻读博士学位）将在本项目的过程中接受创新培训，这将使他们为学术界或工业界的工作做好准备，在那里他们将拥有与同龄人不同的工具和准备。 这将大大提高他们为团队研究环境贡献新的和创新的想法的能力。 除了对专业团体的这种影响之外，推动拟议研究的应用程序对国家安全和扩大残疾人更充分参与社会的能力等不同问题的各个方面产生了明确和直接的影响。

ABSTRACTPI: Michael Kirbyproposal: 0434351This proposal develops new algebro-geometric tools for data analysis, and in particular, for the investigation and understanding of massive data sets arising in computer science, where both the ambient dimension and number of samples may be large. Particular examples of interest to the team include brain-computer interface (BCI) problems and computer-vision and recognition problems. Specifically, the team is relating secant varieties (of arbitrary order) to the determination of optimal projections of data sets; they inquire what order of variety is appropriate and are investigating the relationship between varieties of different orders. Secant varieties also play a role in understanding canonical forms and decompositions of higher-order tensors, which in turn are of fundamental interest in signal-processing applications such as the BCI problem. Algebraic geometry provides a framework for developing algorithms for higher-order tensors. Finally, very recent developments in the Schubert calculus on Grassmannians have dramatically increased the potential applicability of this theory. The investigators are studying how to exploit these ideas to develop efficient algorithms for finding near-optimal projectors subject to constraints. The research, which this team is carrying out, has distinctive impacts on the mathematical and computer science communities. By bringing together groups of researchers from seemingly disparate areas of expertise, it aims for cross-fertilization among these fields. Computer science provides the problems for which the team is developing new algorithms. Conversely, the team expects new mathematical conjectures to arise from the practical problems being addressed by this research. Furthermore, the graduate students (all working towards the Ph.D.) will receive innovative training during the course of this project that will prepare them for jobs in either academia or industry where they will have tools and preparation essentially unlike any of their peers. This should greatly enhance their ability to contribute new and innovative ideas to the team research environment. Beyond this effect on the professional communities, the applications driving the proposed research have clear and immediate impact on aspects of such disparate issues as national security and the broadening of the ability of disabled individuals to participate more fully in society.