Graph-Based Regularization Techniques and Their Applications

科学技术的快速发展迎来了大数据的新时代，需要开发专门的算法来处理大量数据。信号处理和其他相关技术旨在恢复感兴趣的信号或其某些特性；这个目标可以简化为一个优化问题。由于硬件的物理限制，采集数据的大小通常比底层信号的大小小得多，导致信号恢复的病态问题具有无限多个解。正则化技术已经发展到解决这种固有的不适。尽管正则化在低维信号处理中得到了广泛的应用，但它在处理高维数据集方面的应用有限，尤其是那些最好用图表示的数据集，即具有复杂连接的网络。该项目旨在进一步开发基于图形的正则化技术，有可能在数据科学的许多领域彻底改变成像和数据分析技术。该项目旨在为各种信号处理问题开发一个有用的基于图的正则化框架，解决其应用中的主要理论和计算挑战，提供低维正则化技术的新解释，并展示其处理大规模数据集的能力。本研究有三个目标：(1)发展新的基于图的正则化技术，并提供严格的理论保证，以处理更具挑战性的信号处理问题和相关的逆问题；(2)开发高效的数值算法来解决相应的优化问题；(3)在成像应用中进行数值实验，以证明所提出的方法在精度和效率方面的优势。该研究旨在改进数据处理技术，并在医学成像和遥感等各种应用中为数学信号和图像处理注入新的见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The rapid development of science and technology ushers in a new era of big data that requires developing specialized algorithms to process a large amount of data. Signal processing and other related techniques aim to recover signals of interest or some of their properties; this goal can be reduced to an optimization question. Due to physical limitations of hardware, the size of the acquired data is in general much smaller than that of the underlying signal, resulting in an ill-posed problem for signal recovery with infinitely many solutions. Regularization techniques have been developed to address this inherent ill-posedness. Despite being widely applied in low-dimensional signal processing, regularization has seen limited use in processing high-dimensional data sets, especially those best represented by graphs, that is, networks with sophisticated connections. This project aims to further develop graph-based regularization techniques, with potential to revolutionize imaging and data analysis technologies in many areas of data science.This project aims to develop a useful graph-based regularization framework for various signal processing problems, to address major theoretical and computational challenges for its applications, to provide new interpretations of low-dimensional regularization techniques, and to demonstrate its capability for handling large-scale data sets. The research has three objectives: (1) Develop novel graph-based regularization techniques along with rigorous theoretical guarantees to handle the more challenging signal processing problems and related inverse problems; (2) Develop efficient numerical algorithms to solve the corresponding optimization problems; and (3) Conduct numerical experiments in imaging applications to demonstrate the advantages of the proposed approaches in terms of accuracy and efficiency. The research aims to improve data processing techniques and to infuse new insights into mathematical signal and image processing, with a variety of applications such as medical imaging and remote sensing.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.