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

该项目侧重于使用简单锥模型进行矩阵分解，该模型在遥感（特别是高光谱成像），动态频谱访问的射频传感，聚类和主题建模以及社会网络分析等方面具有广泛的应用。该项目的重点是使用凸壳体积最小化标准，为该模型开发健壮且可扩展的计算工具。动机部分来自于最近由主要研究者获得的结果，表明如果采用体积最小化准则，在温和的条件下可以进行唯一因数分解。这些条件比现有方法所要求的条件要现实得多，这表明，只要能够有效地解决相关的优化、健壮性和可伸缩性挑战，就可以实现更具挑战性的场景，甚至新的应用程序领域。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，使研究人员和从业人员能够解决新问题，处理更大的数据集，并提高现有应用程序（如高光谱成像）的性能。在教育方面，该项目将帮助培养一名从事尖端计算工程研究的研究生，并通过高级荣誉项目吸引有才华的本科生，向他们介绍研究和出版的机会。在理论和方法方面，基于体积最小化的矩阵分解的关键方面仍然知之甚少。这项研究将提供一套高性能的计算工具，这些工具植根于对原始体积最小化标准的优缺点的深刻理解，这将带来令人兴奋的发现。该研究将沿着以下协同推力发展：i)体积最小化的鲁棒优化算法；Ii)面向在线体积最小化的可扩展和自适应算法；Iii)验证，使用现有的（例如，高光谱成像）以及有前途的新应用（例如，文档聚类）；iv)体积最小化配方的理论方面，侧重于可识别性和性能界限等基本原理。设计可扩展的体积最小化算法对于涉及快速增加数据量的现代传感和聚类问题非常有意义。从应用的角度来看，频谱感知、信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.