FRG: Collaborative Research: Overcomplete Representations with Incomplete Data: Theory, Algorithms, and Signal Processing Applications

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Driven by accumulated scientific results and recent breakthroughs in sparse representations, recent years have seen an ever-increasing interest in overcomplete expansions with incomplete data---a critical subject requiring close cooperation and exchange of ideas amongst statisticians, mathematicians, and engineers. A number of indicators suggest the appropriateness and timeliness of a Focused Research Group (FRG) involving these three communities as the best means to approach to this high-potential yet challenging research area. In particular, this project follows a comprehensive and vertically integrated research plan for (1) deriving new theoretical results for statistical estimation in the context of overcomplete Gabor time-frequency representations and multiresolution wavelet dictionaries; (2) leveraging these results to develop algorithms tailored for canonical problems in signal and image processing, where practitioners are often faced with missing data or more generally incomplete measurements; and (3) addressing ubiquitous and important cross-cutting applications, including curve fitting as well as audio and color image enhancement. To respond to these pressing scientific needs and prepare the ground for significant developments in the mathematical sciences, the FRG team is exploiting recent results from harmonic analysis and the theory of frames to develop a coherent framework for statistical modeling in the case of overcomplete expansions, including an examination of key open questions such as the impact of the choice of prior coefficient distributions in a Bayesian framework and asymptotic risk bounds for regression when the set of potential predictors is overcomplete. As a definitive first step toward these grand challenges, the team proposes and investigates an innovative common-component model for frame coefficients that recovers currently used methods as special cases but opens up important new avenues for advancement. The FRG team has significant prior experience in multiresolution analysis, computational Bayesian inference, and self-consistency methods for missing data, and hence is also developing and applying state-of-the-art procedures to implement the resulting new algorithms.The Focused Research Group (FRG) project team combines scientists from an established institution (Harvard University) and a young, rapidly growing one (University of Central Florida). The project's research agenda is set to substantially advance the theoretical knowledge and understanding of the applicability of overcomplete representations (a new and important cross-cutting area of mathematics, with many major open questions relating to the area of "compressed sensing" recently featured in the New York Times, The Economist, and elsewhere in the mainstream media) in both statistical and engineering practice. This will ultimately lead to development of more efficient algorithms for signal processing and data analysis in situations where data must be collected at a very low rate (as in the compressed sensing regime described above), or when a portion of available data has been lost or highly contaminated. The latter scenario is particularly salient both for commercial applications (e.g., voice data in the case of cellular communications) as well as military and homeland security concerns (for instance, to recover unobserved data from related sources). Another benefit of the project it its emphasis on close collaboration amongst mathematicians, statisticians, and engineers through a single team, which will lead not only to solution of the specific problems under study, but also to formulations of new important areas of research and their application to the real world. Using support from NSF, the team trains a number of students who are ready to carry out research on the cutting edge of mathematics, statistics and engineering, and holds regular workshops to increase the involvement of new researchers and disseminate results to the wider scientific community.