Learnable Tensor Algebras for Harnessing Implicit Correlations in Multiway Data

大数据已经彻底改变了我们可以解决的各种问题，在商业、科学和医疗保健应用中实现了前所未有的个性化和创新。不断增长的数据量产生了对新方法的迫切需求，以减少存储需求并提取具有代表性的特征以供下游分析。许多数据，如计算机视觉和成像、神经科学、网络（例如，流行病跟踪、网络安全）等领域的数据，都被原生地表示为多路数组或张量。因此，基于张量的方法在降维和特征提取方面变得越来越有吸引力。然而，许多基于张量的方法遭受所谓的“多维诅咒”；也就是说，当应用于多路数据时，基本的数学属性就失效了。张量代数的最新进展克服了这一限制，将张量重新定义为数学运算符，而不是停滞不前的数据数组。该项目将通过学习进一步降低存储成本所需的最佳数学运算，同时提高张量表示的准确性，将这些进步提升到一个新的水平。本项目开发的方法将广泛应用于高影响力的应用，包括精准医疗、气候模拟和工程。所有生成的算法和方法都将以文档完备的开源代码向公众提供。这个项目的重点是开发新的方法，以最大限度地发挥矩阵模拟张量框架的好处-多维框架，保持线性代数性质。与传统的基于矩阵的方法和备选的基于张量的方法相比，这种框架产生了理论和经验上的优势。矩阵拟性源于将张量解释为使用张量-张量乘积相乘的t-线性算子。张量-张量积的选择，由基础张量代数给出，对表示质量至关重要，迄今为止，一直是启发式的。该项目将开发一个统一的优化框架来学习张量代数，并有效地表示具有隐式相关性的多路数据（即未知的先验关系，因此难以启发式捕获）。学习到的张量-张量积将引入算法优势（例如，快速评估和低存储成本），同时保留模拟矩阵框架的理论保证。该项目的主要目标是：(1)通过利用模拟矩阵张量分解和张量-张量积之间的耦合来优化张量代数，(2)通过设计新的非线性张量-张量积来捕捉多线性算法中的非线性，以及(3)使用新的可扩展策略扩展所提出的算法，以增加模拟矩阵张量方法在大规模多路数据应用中的适用性。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。

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.