Building Up the Optimization Algorithmic Infrastructure for Data-Driven Knowledge Discovery and Recovery

研究人员建议研究一些在数据降维、潜在信息提取和隐藏知识发现方面广泛有用的优化模型和算法。重点是涉及非常大的低秩矩阵的问题，包括计算非结构化稠密矩阵的主奇异值分解的基本问题，以及3D图像处理技术在高光谱数据处理和无线视频网络中的应用。总体目标是开发可靠的算法，这些算法比目前使用的算法快得多(在处理大型问题时快一个数量级或更多)。由于许多算法是对经典的增广拉格朗日交替方向法(ALADM)的扩展，该算法最初是针对某些凸规划而设计的，该项目的一部分致力于在更一般的环境下建立ALADM的收敛理论的理论研究。现代技术，如四维CT扫描、卫星遥感和DNA微阵列，正在产生海量且快速可用的数据爆炸。数学和计算技术在帮助及时地、以最少的人为干预的方式从如此庞大的数据集中获得意义方面发挥着至关重要的作用。PI的工作包括研究和设计新的算法来解决几类数学模型，这些模型旨在帮助发现和提取隐藏在大量数据中的最有用的信息。新算法有可能比当今最先进的方法运行得更快，从而为医疗诊断、农业规划、环境监测或生物科学中的遗传研究等众多数据驱动的应用提供更强的处理能力和速度。

The investigator proposes to study a number of optimization models and algorithms that are broadly useful in data dimension reduction, latent information extractionand hidden knowledge discovery. The focus is on problems involving low-rank matrices of very large sizes, including the fundamental problem of computing principal singular value decompositions for unstructured dense matrices, as well as 3D-image processing techniques with applications to hyperspectral data processing and wireless video networks. The overall goal is to develop reliable algorithms that are much faster (by one order of magnitude or more on large problems) than those in use today. Since many of the proposed algorithms are extensions to the classic augmented Lagrangian alternating direction method (ALADM) originally designed for certain convex programs, a part of the project is devoted to a theoretical investigation on establishing a convergence theory for ALADM in more general settings.Modern technologies, such as 4D CT-scans, satellite remote sensing and DNA microarrays, are creating an explosion of data made available in massive quantities and at fast rates. Mathematical and computational techniques play a crucial role in helping make sense out of such massive data sets in a timely fashion and with minimal human interventions. The PI's work involves studying and designing new algorithms for solving several classes of mathematical models designed to help discover and extract the most useful information buried or hidden in large amounts of data. New algorithms have the potential to run much much faster than today's state-of-the-art methods, thus providing more processing power and speed to numerous data-driven applications such as medical diagnoses, agriculture planning, environment surveillance or genetic research in biosciences.