Graph-Based Regularization Techniques and Their Applications

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.