AF: Medium: Collaborative Research: Beyond Sparsity: Refined Measures of Complexity for Linear Algebra

现代数据科学应用程序使用机器学习(包括深度学习)算法来开发现实生活数据中的结构。大多数这些系统的核心是一个被称为线性代数的数学分支的算法。特别是，这些算法中的很大一部分利用了这样一个事实，即现实生活中的数据具有可以使用某些简约线性代数结构来捕获的属性。这个项目研究新的、更强大的线性代数结构和利用这些新结构的算法。鉴于这些算法的根本重要性，该项目产生的想法有望在广泛部署的机器学习系统中实现。该项目的外展部分包括(1)为不同领域的研究人员举办的技术研讨会和(2)为K-12学生举办的外展活动。使用宽度成功地描述了现代数据科学中的各种问题。例如，矩阵的秩是最常见的宽度之一，在许多应用程序中几乎无处不在地使用。该项目极大地扩大了对最近提出的几个宽度的理解，并充分挖掘了它们取得积极实际成果的潜力。此外，它还有助于最近线性代数、机器学习和编码理论中超越最坏情况分析的工作。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。

Modern data science applications exploit structure in real life data using machine learning (including deep learning) algorithms. At the core of most of these systems are algorithms for a branch of mathematics called linear algebra. In particular, a large portion of these algorithms utilize the fact that real life data has properties that can be captured using certain parsimonious linear algebraic structures. This project studies new, more powerful linear algebraic structures and algorithms that exploit these new structures. Given the fundamental importance of these algorithms, ideas generated from this project are expected to be implemented in widely deployed machine learning systems. The outreach component of this project involves (1) a technical workshop for researchers from diverse areas and (2) outreach events for K-12 students.A variety of problems in modern data science have been successfully characterized using a width. For example, one of the most common widths, the rank of a matrix, has a near-ubiquitous use across many applications. This project significantly expands the understanding of several recently proposed widths and extracts their full potential for positive practical outcomes. Furthermore, it contributes to the recently growing work on beyond worst-case analysis in linear algebra, machine learning and coding theory.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.