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

研究者提出了一些优化模型和算法，这些模型和算法在数据降维、潜在信息提取和隐藏知识发现方面具有广泛的应用。重点是涉及非常大的尺寸的低秩矩阵的问题，包括计算非结构化密集矩阵的主要奇异值分解的基本问题，以及应用于高光谱数据处理和无线视频网络的3D图像处理技术。 总体目标是开发可靠的算法，比目前使用的算法快得多（在大型问题上快一个数量级或更多）。由于所提出的许多算法都是对经典的增广拉格朗日交替方向法（ALADM）的扩展，该方法最初是为某些凸规划设计的，因此该项目的一部分致力于在更一般的情况下建立ALADM的收敛理论的理论研究。现代技术，如4D CT扫描，卫星遥感和DNA微阵列，正在创造一个数据的爆炸性增长，这些数据以巨大的数量和快速的速度提供。数学和计算技术在帮助及时了解如此庞大的数据集方面发挥着至关重要的作用，并以最少的人为干预。PI的工作包括研究和设计用于解决几类数学模型的新算法，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快得多，从而为许多数据驱动的应用提供更多的处理能力和速度，例如医疗诊断，农业规划，环境监测或生物科学中的遗传研究。

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.