Future scientific and technological progress will depend heavily on the generation of new information technology capabilities and novel methods from signal and image processing to deal with today's massive volumes of data. A research effort is proposed to create mathematical concepts and computational methods to address some of the key challenges in this important area. In particular, the PI will focus on the areas of imaging, high-dimensional data analysis, machine learning, and information theory. The project uses tools from computational harmonic analysis, operator theory, random matrix theory, and optimization yielding efficient numerical algorithms with rigorously-established properties under carefully stated conditions. The payoffs for society at large are many, including new information technology capabilities, improved methods for signal- and image processing, as well as better understanding of data mining tools for Big Data.
Two concrete topics of this research effort are:(i) Fast and reliable algorithms of non-convex problems: When dealing with massive data sets, many tasks involve the use of a heuristic algorithm to solve a non-convex optimization problem. Often these heuristic algorithms get stuck in local minima, that are far away from the global minimum. We will develop fast numerical algorithms that come with theoretical performance guarantees for a range of important data analysis tasks; (ii) Efficient algorithms for heterogenous and high-dimensional data: Existing methods for high-dimensional data are often computationally rather expensive and rely on stationarity and homogeneity of the data, thus limiting their use for massive, heterogenous data sets. The PI will derive a framework of computationally efficient methods for properly fusing and efficiently processing heterogeneous, high-dimensional data.
