The investigator proposes to study a number of optimization models and algorithms that are broadly useful in data dimension reduction, latent information extraction and 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.
