Robust and Scalable Volume Minimization-based Matrix Factorization for Sensing and Clustering

该项目侧重于使用单纯锥模型的矩阵分解，该模型在遥感(特别是高光谱成像)、用于动态频谱访问的射频传感、集群和主题建模以及社会网络分析等方面有着广泛的应用。该项目的重点是使用凸壳体积最小化标准，为该模型提供健壮和可扩展的计算工具。其动机部分来自主要研究者最近得到的一个结果，该结果表明，如果采用体积最小化标准，在温和的条件下唯一的因式分解是可能的。这些条件远比现有方法所要求的条件更现实，这表明，如果能够有效地解决相关的优化、健壮性和可伸缩性挑战，更具挑战性的场景甚至新的应用程序域是可以实现的。这项研究将为这些令人兴奋的发展提供计算基础。高性能体积最小化软件将公开发布，使研究人员和从业者能够解决新问题，处理更大的数据集，并提高现有应用程序的性能，如高光谱成像。在教育方面，该项目将帮助培养尖端计算工程研究方面的研究生，并将通过高级荣誉项目帮助吸引有才华的本科生，向他们介绍研究和出版机会。在理论和方法方面，基于体积最小化的矩阵分解的关键问题仍然知之甚少。这项研究将提供一套高性能的计算工具，它植根于对原始体积最小化标准的优势和劣势的深刻理解，该标准有望带来令人兴奋的发现。研究将沿着以下协同方向发展：i)体积最小化的稳健优化算法；ii)面向在线体积最小化的可扩展和自适应算法；iii)使用现有的(例如，高光谱成像)以及有希望的新的(例如，文档聚类)应用进行验证；以及iv)体积最小化公式的理论方面，重点关注诸如可识别性和性能界限之类的基本原理。设计可伸缩的体积最小化算法对于涉及快速增长的数据量的现代传感和聚类问题具有重要意义。从应用的角度来看，频谱感知、通道识别和文档聚类的体积最小化是全新的和具有挑战性的。

This project focuses on matrix factorization using a simplicial cone model, which has a wide variety of applications in remote sensing (particularly hyperspectral imaging), radio frequency sensing for dynamic spectrum access, clustering and topic modeling, and social network analysis, to name a few. The project focuses on robust and scalable computational tools for this model, using a convex hull volume minimization criterion. The motivation partially comes from a result that was recently obtained by the principal investigators, showing that unique factorization is possible under mild conditions if one adopts the volume minimization criterion. These conditions are far more realistic than those required by existing approaches, suggesting that more challenging scenarios and even new application domains are within reach if only related optimization, robustness, and scalability challenges can be effectively addressed. This research will provide the computational underpinnings of these exciting developments. High-performance volume minimization software will be publicly released to enable researchers and practitioners to tackle new problems, handle much larger datasets, and boost performance in existing applications like hyperspectral imaging. On the education front, the project will help train a graduate student in cutting-edge computational engineering research, and will also help engage talented undergraduates through senior honors projects, introducing them to research and publication opportunities. In terms of theory and methods, key aspects of volume minimization-based matrix factorization are still poorly understood. The research will provide a set of high-performance computational tools rooted in deep understanding of the strengths and weaknesses of the original volume minimization criterion which promises exciting discoveries. The research will evolve along the following synergistic thrusts: i) robust optimization algorithms for volume minimization; ii) scalable and adaptive algorithms towards online volume minimization; iii) validation, using existing (e.g., hyperspectral imaging) as well as promising new (e.g., document clustering) applications; and iv) theoretical aspects of the volume minimization formulation, focusing on fundamentals such as identifiability and performance bounds. Devising scalable volume minimization algorithms makes a lot of sense for modern sensing and clustering problems which involve rapidly increasing amounts of data. From an applications point of view, volume minimization for spectrum sensing, channel identification, and document clustering are completely new and challenging.