Multilinear Subspace Techniques for Learning and Recovery through Tensor-Tensor Decompositions

随着数字设备每天不断产生大量数据，数据科学家的目标是分析这些数据，努力发现人类调查无法识别的模式。模式发现（或模式分析）通常涉及将数据的维数降低到数据科学家可以管理的程度的算法。现代数据科学的一个主要目标是根据嵌入模式的数据确定最适合模式发现的算法。该项目旨在研究一组新的数据发现算法，这些算法特别适合以多维格式分析数据。这项研究的结果可以提供新的技术来帮助分析和解释现有的医学成像和信号数据，以帮助诊断和预防医学异常；允许农民解释和分析多/高光谱数据，以提高作物产量；分析和解释生物识别数据和人体步态，以增加个人和国家安全；并提供新的方法来分析网络（从社会到点对点），仅举几例。此外，这项研究将培养一个由研究生和本科生组成的多学科团队，结合计算科学、数学科学和统计科学的专业知识，成为数据科学和模式发现的下一代领导者。该项目由稳健情报（RI）和刺激竞争研究的既定计划（EPSCoR）共同资助。当前项目的技术目标有两个：1)研究新的多线性子空间学习技术，以发现多维数据中的隐藏模式；2)研究用于数据重建和解释的多线性最佳样本和恢复技术的新方法。为此，研究人员将探索一种根本不同的方法来分解多路阵列，通常称为张量。该方法基于最近开发的张量-张量分解，视为圆卷积，其中类似于其线性代数对立物，分解结果是三个张量的乘积，每个张量都具有定义结构。通过这些分解，研究小组将对线性子空间学习的许多变体（及其数学关系）开发新的多线性扩展，其中包括多维缩放、局部保留投影、独立分量分析、典型相关分析和偏最小二乘，以及通过多线性扩展到压缩感知、稀疏逼近、还有稀疏编码。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。

As digital devices continue to produce massive amounts of data on a daily basis, data scientist aim to analyze this data in an effort to discover patterns not discernible by human investigation. Pattern discovery (or pattern analysis) often involves algorithms that reduce the dimensionality of the data to something manageable for data scientists to work with. One major goal of modern data science is to determine which algorithm is best suited for pattern discovery based on the data to which the patterns are embedded. This project aims to investigate a new family of algorithms for data discovery that are particularly well suited for data being analyzed in multi-dimensional formats. The results of this research could provide novel techniques to aid in the analysis and interpretation of existing medical imaging and signal data to help diagnose and preempt medical anomalies; allow farmers to interpret and analyze multi/hyper-spectral data to increase crop yield; analyze and interpret biometric data and human gaits for increased personal and national security; and provide new approaches to the analysis of networks (from social to peer-to-peer), to name but a few. Furthermore, this research will educate a multidisciplinary team of graduate and undergraduate students combining expertise from the computing sciences, mathematical sciences, and statistical sciences to be next generation leaders in data science and pattern discovery. This project is jointly funded by Robust Intelligence (RI) and the Established Program to Stimulate Competitive Research (EPSCoR).The technical goals of the current project are twofold: 1) investigate new multilinear subspace learning techniques to uncover hidden patterns in multidimensional data and 2) investigate new approaches to multilinear optimal sample and recovery techniques for data reconstruction and interpretation. Toward this end the investigators will explore a fundamentally different approach to decompositions of multi-way arrays, commonly referred to as tensors. The approach is based on a recently developed tensor-tensor decomposition viewed as circular convolution where, similar to their linear algebraic counterparts, the decompositions result in the product of three tensors, each with a defining structure. Through these decompositions, the investigative team will develop new multilinear extensions to the many variations of linear subspace learning (and their mathematical relationship), a subset of which include multidimensional scaling, locally preserving projections, independent component analysis, canonical correlation analysis, and partial least squares as well as novel advances to optimal sample and recovery through multilinear extensions to compressed sensing, sparse approximations, and sparse coding.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.