Tensors, Topics, Truth, and Time: Methods for Real Tensor Applications

随着最近涉及大规模数据的应用程序的激增，迫切需要开发高效、健壮和实用的数据分析方法。随着越来越多的应用程序具有多模式数据(来自许多不同来源和不同类型的数据，通常具有时间成分)，需要数学开发来处理和理解这些数据。这类研究的核心数学对象是张量，张量是一种多维数组，可以看作是数学矩阵这一常见概念的代数扩展。张量的数学近年来受到了广泛的关注，然而，在对这些对象的科学理解以及它们在现代数据分析技术中的应用方面，仍然存在许多关键的缺陷。该项目的重点是开发在计算上可行的方法来检测这种张量数据中的模式，以及可用于压缩的张量的几何性质。该团队将与加州无辜项目(CIP)合作，该项目是一个非营利性组织，其目标是释放被判有罪的无辜者。涉及的数据是囚犯信件和案件档案，所有这些都是高度多态的，因为它们包括例如法庭文件、面谈证词、法医数据、图像等。在这方面的目标将是促进CIP用来决定哪些案件可能成功、它们有哪些共同点以及哪些人群可能需要更多关注的评估程序。与CIP的伙伴关系不仅将作为一种直接的社会影响的手段，而且还将作为一种反馈机制来测试和验证所开发的数学方法。第一个推力集中在方法上，在不可能展开的情况下，在在线环境中检测张量数据中的模式，并允许主题结构。第二个重点是降维，开发作用于张量和映射到张量的保持几何的降维映射，以及利用这种映射的相关重要方法。第一个推力将在几个方面超越现有研究。首先，它将在动态应用程序中提供大大改进的话题检测。其次，它将在随机在线设置中开发可证明的特征收敛。第三，它将使用深度模型提供改进的主题结构。第二个重点是张量降维的数学，并将提供可证明的保证，以及相关算法的分析。对于真正的张量数据，这样的实用技术和理解还不存在。因此，拟议的研究计划将进一步加深对张量几何的数学理解，同时也提供了可用于任何需要分析多峰数据的领域的实用方法。这些成果向社会的转变将通过PI和非营利组织之间的联系来促进，包括加州天真项目。该项目还包括一个新颖的外展和教育部分，包括将高中生计划、社区变革计划、教师和未来的教师整合到全年的暑期研讨会和活动中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

With the recent surge of applications involving large-scale data comes a critical need to develop efficient, robust, and practical methods for data analysis. With more and more applications having multi-modal data (data coming from many distinct sources and of different types, often having a temporal component), the need for mathematical developments to handle and understand this data is critical. The key mathematical object at the heart of such study is the tensor — a multi-dimensional array that can be viewed as an algebraic extension of the common notion of a mathematical matrix. The mathematics of tensors has received a lot of recent attention, however, there are still many key lacunae in the scientific understanding of these objects as well as their use in modern data analytic techniques. The project focuses on the development of computationally feasible methods to detect patterns within such tensor data as well as the geometric properties of tensors that can be used for compression. The team will partner with the California Innocence Project (CIP), a nonprofit whose goal is to free innocent persons who have been convicted of a crime. The data involved are inmate letters and case files, all of which are highly multi-modal since they include, for example, court documents, interview testimonials, forensic data, images, and more. The goals in this context will be to facilitate the assessment procedure that CIP uses to decide what cases are likely to be successful, what commonalities they share, and what populations may need more attention. The partnership with CIP will serve not only as a means of direct societal impact but also as a feedback mechanism to test and validate the developed mathematical approaches.We focus on two technical thrusts. The first thrust is centered around methods to detect patterns in tensor data without impossible unfoldings, in an online setting, and allowing for topic structures. The second thrust focuses on dimension reduction, developing geometry preserving reduction maps that act on tensors and map to tensors, along with related important methods that utilize such maps. The first thrust will go beyond existing research in several ways. First, it will provide much improved topic detection in dynamic applications. Second, it will develop provable convergence of features in the stochastic online setting. Third, it will offer improved topic structures using a deep model. The second thrust focuses on the mathematics of tensor dimension reduction and will provide provable guarantees for such, along with analysis of the related algorithms. Such practical techniques and understanding simply do not yet exist for true tensor data. The proposed research program will therefore further mathematical understanding of tensor geometries while also providing practical approaches that can be used in any field needing to analyze multi-modal data. The transition of these results to society will be facilitated through connections between the PI and nonprofits, including the California Innocence Project. The project also includes a novel outreach and educational component, including integration of high school student programs, community change programs, teachers, and future teachers in summer workshops and events throughout the year.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.