Greedy Approximation in Banach Spaces and Compressed Sensing

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Compressed sensing is a recent theory that aims to approximate a signal using only a small number of measurements. It is motivated by the fact that many real-world signals can be well-approximated by or identified with sparse signals.The proposed research will utilize techniques based on nonlinear sparse representations with respect to redundant systems (dictionaries). The investigator's preliminary results show that, although the prior development of compressed sensing has occurred in Hilbert spaces, substantial improvement can be made in the construction of efficient algorithms for exact recovery of sparse signals when we consider the problem in a Banach space instead. To widen the search for efficient algorithms from Hilbert spaces to Banach spaces is the main fundamentally new idea of this proposal. We intend to study properties of specific methods of recovery that belong to a family of greedy algorithms. These methods allow us to build sparse representations economically. The proposed research will develop practical, implementable algorithms that are provably efficient with respect to convergence, rate of convergence, and exact recovery.For scientists and other analysts, the great challenges of this decade involve designing systems to efficiently analyze data and extract essential information from large data sets. It has a myriad of existing and potential applications in both the defense and civilian sectors. For example, managing large data bases, such as security data bases obtained through surveillance, requires classification and compression of the data in order to facilitate the extraction of significant features or specific information. More generally, the problem of compression and denoising of large data sets arises naturally in biology, medicine, and many other fields. The scientific discipline that studies this aspect of data processing is called 'compressed sensing.' The goal of the proposed research is to build a fundamental mathematical theory that will significantly increase our ability to process (compress, denoise, etc.) large data sets. The main technique that will be used to achieve this goal is based on nonlinear sparse representations arising from greedy approximation methods (greedy algorithms), which allow us to build sparse representations economically. It is the goal of this proposal to utilize fundamental concepts in analysis and applied mathematics to clearly define and quantify the data processing challenge and design new, more efficient techniques (greedy algorithms) to address it. We believe that scientists and other analysts will be able to implement our results to find essential information in large data sets more quickly and more accurately than before.