Learnable Tensor Algebras for Harnessing Implicit Correlations in Multiway Data

用于利用多路数据中隐式相关性的可学习张量代数

• 批准号: 2309751
• 负责人: Elizabeth Newman
• 金额: $ 23万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309751&HistoricalAwards=false
• 关键词: Learnable; Tensor; Algebras; Harnessing; Implicit

Big data has revolutionized the kinds of problems we can tackle, enabling unprecedented personalization and innovation across commercial, scientific, and healthcare applications. The ever-growing amount of data has created a pressing need for new methodologies to reduce storage demands and extract representative features for downstream analysis. Many data, such as those arising in computer vision and imaging, neuroscience, networks (e.g., epidemic tracking, cyber security), and more, are natively represented as multiway arrays, or tensors. As a result, tensor-based approaches have become increasingly attractive for dimensionality reduction and feature extraction. However, many tensor-based approaches suffer from a so-called “curse of multidimensionality;” that is, that fundamental mathematical properties break down when applied to multiway data. Recent advances in tensor algebra have overcome this limitation by reframing tensors as mathematical operators rather than stagnant arrays of data. This project will take these advancements to the next level by learning the optimal mathematical operations required to drive down storage costs further while increasing the accuracy of tensor representations. The methods developed in this project will be useful for a wide range of high-impact applications, including precision medicine, climate simulations, and engineering. All algorithms and methods produced will be made available to the public in well-documented, open-source code. This project focuses on developing new methods to maximize the benefits of matrix-mimetic tensor frameworks- multidimensional frameworks that preserve linear algebraic properties. Such frameworks yield theoretical and empirical advantages over traditional matrix-based approaches and alternative tensor-based approaches. The matrix mimeticity arises from interpreting tensors as t-linear operators that multiply using tensor-tensor products. The choice of tensor-tensor product, given by an underlying tensor algebra, is crucial to representation quality, and thus far, has been made heuristically. This project will develop a unifying optimization framework to learn tensor algebras and efficiently represent multiway data with implicit correlations (i.e., relationships unknown a priori and thus challenging to capture heuristically). The learned tensor-tensor products will introduce algorithmic advantages (e.g., fast evaluations and low storage costs) while preserving theoretical guarantees of the matrix-mimetic framework. The main thrusts of this project are (1) to optimize tensor algebras by exploiting the coupling between matrix-mimetic tensor factorizations and tensor-tensor products, (2) to capture nonlinearity in multilinear algorithms by designing novel nonlinear tensor-tensor products, and (3) to extend the proposed algorithms using new, scalable strategies to increase the applicability of matrix-mimetic tensor approaches to massive multiway data applications.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.

大数据彻底改变了我们可以解决的各种问题，从而使商业，科学和医疗保健应用中的前所未有的个性化和创新。不断增长的数据量引起了新方法的迫切需求，以减少存储需求并提取代表性特征以进行下游分析。许多数据，例如在计算机视觉和成像中产生的数据，神经科学，网络（例如，流行性跟踪，网络安全）等是多道路阵列或张量。结果，基于张量的方法对于降低维度和特征提取方面变得越来越有吸引力。但是，许多基于张量的方法遭受了所谓的“多维性诅咒”；也就是说，将基本数学属性分解为多路数据。张量代数的最新进展通过将张量作为数学运算符而不是停滞的数据来克服了这一限制。该项目将通过学习进一步降低存储成本所需的最佳数学操作，同时提高张量表示的准确性，从而将这些进步提升到一个新的水平。该项目中开发的方法将用于广泛的高影响应用程序，包括精密医学，气候模拟和工程。所有生产的算法和方法将以有据可查的开源代码向公众提供。该项目着重于开发新方法，以最大程度地提高矩阵模拟张量框架的好处，即保留线性代数属性的多维框架。这种框架比传统基于矩阵的方法和替代张量的方法具有理论和经验优势。矩阵模拟性来自将张量解释为使用张量张量产物繁殖的T线性操作员。通过基础张量代数给出的张量张量产品的选择对于表示质量至关重要，到目前为止，已经启发了。该项目将开发一个统一的优化框架，以学习张量代数，并有效地表示具有隐式相关性的多路数据（即，关系未知的先验性，因此可以捕获启示性的挑战）。学识渊博的张量调整产品将引入算法优势（例如，快速评估和低存储成本），同时保留矩阵模拟框架的理论保证。 The main thrusts of this project are (1) to optimize tensor algebras by exploiting the coupling between matrix-mimetic tensor factorizations and tensor-tensor products, (2) to capture nonlinearity in multilinear algorithms by designing novel nonlinear tensor-tensor products, and (3) to extend the proposed algorithms using new, scalable strategies to increase the applicability of矩阵模拟的张量方法用于大规模多路数据应用。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估，以诚实的支持。