CCSS: Block-term Tensor Tools for Multi-aspect Sensing and Analysis

多方面数据普遍存在于传感、通信、信号处理等领域。例如，遥感中的高光谱图像具有两个空间方面和一个光谱方面；无线通信中的频谱动力学数据包括频率、时间和空间方面。作为矩阵的高阶扩展，张量模型被认为是多方面感知和分析不可或缺的工具。本课题研究的是最近出现的一种张量分解模型，即多线性秩块分解模型(L，L，1)(即LL1模型)。这个模型与遥感、医学成像、化学计量学和无线通信等许多核心应用的基本物理很好地结合在一起，从而为这些任务提供了强有力的理论保证。然而，LL1模型相对较新，因此与其理论和方法有关的许多挑战(例如，可伸缩性、健壮性和缺失值可恢复性)在很大程度上仍是未知领域。提升工程应用程序性能的诱人承诺几乎没有得到充实。该项目将极大地促进对LL1模型计算和分析方面的理解-导致一系列有理论支持的多方面数据采集和处理算法。除了工程学，发展的理论和方法还将广泛应用于相关领域，如生态学、生物学和食品科学，这些领域经常出现多方面的数据。该项目还将为本科生提供优化、线性代数和真实数据获取方面的培训机会。为了充分利用LL1张量模型的力量，该项目将解决LL1分解理论和方法中的一些关键挑战。具体地说，第一个推力将设计可扩展的LL1算法，它可以灵活地结合各种先验信息作为约束和正则化；第二个推力将为存在严重离群点的LL1计算开发模型识别保证和算法；第三个推力将开发用于恢复压缩/下采样LL1张量的理论和算法；以及最后一个推力将所提出的方法应用于实际工程问题，如高光谱分解、荧光数据分析、高光谱超分辨率和光谱制图。随着多方面数据的迅速增长，拟议的计算和分析工具动机良好。由于LL1张量分解的非凸非光滑性质和LL1模型™S固有的病态多线性结构，与LL1张量分解相关的优化问题在理论和实践中都是困难的。离群点稳健的LL1分解和压缩/下采样的LL1张量恢复提出了令人兴奋的研究问题，这些问题位于信号处理和数据分析的核心。该项目开发的解决方案将为这些具有挑战性的开放问题提供新的见解和有效的工具，这些问题必然会使广泛的现实世界应用程序受益。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。

Multi-aspect data arise ubiquitously in sensing, communications, and signal processing. For example, a hyperspectral image in remote sensing has two spatial aspects and one spectral aspect; spectrum dynamics data in wireless communications comprise frequency, time, and space aspects. As higher-order extensions of matrices, tensor models are considered indispensable tools for multi-aspect sensing and analytics. This project is concerned with a recently emerged tensor decomposition model, namely, the block-term decomposition model in multilinear rank-(L,L,1) terms (i.e., the LL1 model). This model is well-aligned to the underlying physics of a number of core applications in remote sensing, medical imaging, chemometrics, and wireless communications, thereby offering strong theoretical guarantees for these tasks. However, the LL1 model is relatively new, and thus many challenges pertaining to its theory and methods (e.g., scalability, robustness, and missing value recoverability) are still largely uncharted territories. The appealing promises for boosting performance of engineering applications have been barely fleshed out. This project will significantly advance the understanding to the computational and analytical aspects of the LL1 model --- leading to a series of theory-backed refreshing multi-aspect data acquisition and processing algorithms. Beyond engineering, the developed theory and methods will also be broadly applicable in related domains such as ecology, biology and food science, where multi-aspect data frequently come up. The project will also offer opportunities for training undergraduate students in optimization, linear algebra, and real-data acquisition.Towards fully capitalizing the power of the LL1 tensor model, this project will address a number of critical challenges in LL1 decomposition theory and methods. Specifically, the first thrust will design scalable LL1 algorithms that can flexibly incorporate a large variety of prior information as constraints and regularization; the second thrust will develop model identification guarantees and algorithms for LL1 computations in the presence of gross outliers; the third thrust will develop theory and algorithms for recovering compressed/downsampled LL1 tensors; and the last thrust applies the proposed approaches onto real-world engineering problems such as hyperspectral unmixing, fluorescence data analysis, hyperspectral super-resolution, and spectrum cartography. The proposed computational and analytical tools are well-motivated in wake of the rapid growth of multi-aspect data. Optimization problems associated with LL1 tensor decomposition are hard in both theory and practice, due to their nonconvex nonsmooth nature and the LL1 model’s inherent ill-conditioned multilinear structure. Outlier-robust LL1 decomposition and compressed/downsampled LL1 tensor recovery pose exciting research questions that reside at the core of signal processing and data analytics. The solutions developed in this project will offer new insights and effective tools for these challenging open problems, which are bound to benefit a broad range of real-world 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.