Collaborative Research: OAC Core: Robust, Scalable, and Practical Low Rank Approximation

社会的几乎所有方面都受到近年来以更快速度产生的数据的影响。来自实验、观察和模拟的数据不仅存在于更经典的科学和工程领域，而且还存在于许多其他领域，例如企业跟踪消费者行为的越来越多方面，以及社交网络捕获关于人与其行为和互动之间的关系的大量信息。强烈需要将一组数据提取成较小的表示，该表示将有用的信息与噪声分开并捕捉最重要的趋势、模式，和底层关系。 这样的表示可用于直接解释隐藏模式或作为简化其他数据分析任务的手段。 此项目通过研究线性代数中称为低阶近似的概念来解决这些挑战。 该项目开发了忠实地提取数据集中有意义的信息的技术。 算法还被设计为利用高性能计算机，以便分析师能够更快地获得结果并解决更大的问题。 项目中的整体努力有望弥合能够有效处理超大规模问题的算法与数据分析师转换能力之间的差距在这个项目中研究的矩阵和张量低阶近似是许多科学和工程应用中的基本工具。对低阶近似施加约束可以对许多关键问题进行建模，而设计可扩展的算法可以实现远远超出经典科学和工程学科的新应用。特别是，具有非负数据值的数学模型比比皆是，施加非负约束允许更准确和可解释的模型。这些约束的变体可以设计为反映现实生活中数据分析问题的其他特征。该项目的主要目标是(1)开发用于评估用于确定等级和模型的计算低阶近似的健壮技术，(2)开发可伸缩的并行算法，用于在当今极大规模的机器上进行大规模且健壮的低阶近似，以及(3)为最终用户提供大规模计算和分析解决方案所需的实用工具。典型的数据和应用科学家使用Python或MatLab迭代计算、可视化和评估解决方案，并且他们仅限于具有可行的内存和计算要求的小数据集。虽然存在高性能的算法和实现，但如果最终用户不能依赖结果的健壮性和普适性，他们就不会利用这些工具。该项目旨在缩小这一差距，开发一个端到端系统，为数据分析工作流程的所有步骤提供可扩展的解决方案。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。

Nearly all aspects of society are affected by data being produced at a faster rate in recent years. The data from experiments, observations, and simulations are not only in more classical science and engineering domains but also in numerous other areas such as businesses tracking more and more facets of consumer behavior, and social networking capturing vast amounts of information on the relationships between people and their actions and interactions. There is a strong need to distill a set of data into a smaller representation that separates useful information from noise and captures the most important trends, patterns, and underlying relationships.  Such a representation can be used for direct interpretation of hidden patterns or as a means of simplifying other data analytic tasks.  This project addresses these challenges by studying a concept from linear algebra called low rank approximation.  The project develops techniques that faithfully distill the meaningful information within a data set.  The algorithms are also designed to exploit high-performance computers so that analysts can get results more quickly and tackle larger problems.  The overall effort in the project is expected to close the gap between algorithms that can effectively handle very large-scale problems and the data analyst’s ability to convert raw input into meaningful representations and actionable insight.The matrix and tensor low rank approximations being studied in this project serve as foundational tools in numerous science and engineering applications. Imposing constraints on the low rank approximations enables the modeling of many key problems, and designing scalable algorithms enables new applications that reach far beyond classical science and engineering disciplines. In particular, mathematical models with nonnegative data values abound, and imposing nonnegative constraints allows for more accurate and interpretable models. Variants of these constraints can be designed to reflect additional characteristics of real-life data analytics problems. The primary goals of this project are (1) to develop robust techniques for evaluating computed low rank approximations for rank and model determination, (2) to develop scalable parallel algorithms for large and robust low rank approximations on today’s extreme-scale machines, and (3) to provide end users the practical tools required to compute and analyze solutions at scale. Typical data and application scientists use Python or Matlab to iteratively compute, visualize, and evaluate solutions, and they are limited to small data sets with feasible memory and computational requirements. While high-performance algorithms and implementations exist, end users would not leverage these tools if they cannot rely on the robustness and generalizability of the results. This project aims to close this gap, developing an end-to-end system with scalable solutions for all steps of the data analytics workflow.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.