This research project examines the theoretical, algorithmic, and computational issues that arise in compressed sensing (CS) and signal processing problems where there is a need to compute solutions to problems in which the solution vector has many zeros. In addition to the exciting compressed sensing area, this research will benefit numerous signal processing applications where the sparsity constraint on the solution vector naturally arises. Brain imaging techniques such as Magnetoencephalography (MEG) and Electroencephalography (EEG) are currently important examples. Sparse communication channels with large delay spread, high resolution spectral analysis, and direction of arrival estimation, are other important examples. An effective solution to this problem will have significant impact, by providing new and valuable tools to the practicing signal processing engineer. In addition, the tools will be of interest to researchers in cognitive science, neuroscience, and machine learning where sparsity issues naturally arise, such as sparse coding of signals in the brain or learning from data which is often assumed to lie on a low dimensional manifold.
This project provides a comprehensive and tighter integration of the compressed sensing field and multi-user information theory. This makes it possible to utilize the rich results available in network information theory which have been successfully applied to the implementation of communication systems. The theoretical tools necessary to enable this integration are being developed by the investigators. This research enables significant advances in both theory and practice in the CS field. The information theoretic insights are leveraged to provide insights on performance limits and guidance on practical CS-based system design. The implementation experience gained from communication systems will be translated to practical algorithm development and efficient CS-based system design.
This research project examined the problem of sparse signal recovery and compressive sensing. The objectives of the work were to further the theoretical understanding of the sparse signal recovery problem, advance the algorithmic state of the art, and expand the application domain for the framework. More specifically, the research objectives were 1. Develop insight into the fundamental limits of signal recovery algorithms. These limits are algorithm agnostic and provide important bounds on what is potentially achievable. 2. Develop novel algorithms that have state of the art performance and extensible to deal with a variety of environments. 3. Explore the viability of the methods in a variety of important applications. The main results are summarized below and more details can be found in our publications that have appeared in top tier conferences and journals. 1. To address the theoretical question of performance limits, network information theory was used in a novel manner to shed light on what is possible for the important problem of multiple measurement vectors, block sparsity, and arbitrarily distributed random measurement matrices. The connection between the sparse signal recovery problem and the multiple access comunication problem is a key step in this study. 2. For the algorithmic work, novel algorithms were developed using the sparse bayesian learning framework to deal with block sparsity in its most general form. A key feature of the algorithms developed is the flexibility with which structure can be readily incorporated. 3. With regards to applications, novel algorithms for telemonitoring of biomedical (EEG, ECG) signals as well as novel approaches to the deconvolution and speech modeling problem were developed. This work will have impact on many fronts. The performance limits developed using network information theory are in a very general setting and should siginificantly advance the state of the art on the theoretical front. The algorithms developed using sparse Bayesian theory are very flexible and should be applicable in a variety of scenarios. The biomedical and deconvolution applications studied benefit from the new and promising techniques. More generally, the results developed as part of this research should impact many areas such as communications, biomedical signal processing, speech modeling, among others. The technology it enables in the areas of communications and medicine will benefit all members of society. The project has enabled the training of several (4) students in the forefronts of information theory and signal processing. Two students working on this project successfully defended their Ph.D. thesis. These skills are very relevant to industry and for navigating the information age. The students graduating are joining industry and using the skills learnt to develop future products. As a result, the algorithms developed are likely to find their way into future commercial products.
