NSF Award
2309751
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. <br/><br/>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.<br/><br/>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.
