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态数据（来自许多不同来源和不同类型的数据，通常具有时间组件），对处理和理解这些数据的数学开发的需求至关重要。这种研究核心的关键数学对象是张量——一个多维数组，可以被看作是数学矩阵一般概念的代数扩展。近年来，张量的数学研究受到了广泛的关注，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空白。该项目侧重于开发计算上可行的方法来检测这些张量数据中的模式，以及可用于压缩的张量的几何属性。该小组将与加州无罪项目（CIP）合作，这是一个非营利组织，其目标是释放被判有罪的无辜人员。所涉及的数据是囚犯信件和案件档案，所有这些都是高度多模式的，因为它们包括，例如，法庭文件，面谈证词，法医数据，图像等等。在这种情况下，目标将是促进CIP使用的评估程序，以确定哪些案例可能成功，它们有什么共同点，以及哪些人群可能需要更多关注。与CIP的伙伴关系不仅将作为一种直接的社会影响手段，而且还将作为一种反馈机制来测试和验证已开发的数学方法。我们专注于两个技术重点。第一个重点是在不可能展开的情况下检测张量数据模式的方法，在在线设置中，并允许主题结构。第二个重点是降维，发展保留几何的降维图，这些降维图作用于张量并映射到张量，以及利用这些图的相关重要方法。第一个推动力将在几个方面超越现有的研究。首先，它将在动态应用程序中提供大大改进的主题检测。其次，它将在随机在线设置中发展可证明的特征收敛性。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，这种实用的技术和理解还不存在。因此，拟议的研究计划将进一步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模态数据的领域的实用方法。通过PI和非营利组织（包括加州无罪项目）之间的联系，将促进这些成果向社会的转化。该项目还包括一个新颖的推广和教育组成部分，包括在全年的夏季研讨会和活动中整合高中学生计划、社区变革计划、教师和未来教师。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.