Banach Spaces with Applications to Compressed Sensing and Greedy Convergence

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This project will use Banach space theory to investigate some problems that are motivated by applications to data compression. It will improve the recent explicit constructions of matrices with the Restricted Isometry Property (RIP) so as to match more closely the optimal efficiency of probabilistic constructions. RIP matrices are used in compressed sensing to reconstruct a sparse signal by a measurements vector of much smaller dimension than that of the signal. A vector contained in a Banach space can be approximated by finite expressions involving the elements of a basis or a redundant dictionary such as a frame. Typically these approximants are selected by a greedy algorithm such as the X greedy Algorithm or the Thresholding Greedy Algorithm. Convergence results for these algorithms depend on geometrical properties of the Banach space such as uniform smoothness. They also depend on unconditionality properties of the dictionary. An important related open problem in Banach space theory to be resolved is to show that the family of constants (Elton constants) corresponding to the nonlinear projection of a vector onto sets of large coefficients is uniformly bounded. Closely connected to this is the quantization problem of replacing arbitrary real coefficients by a finite alphabet of coefficients. The existence of dictionaries with good quantization properties will be related to the geometrical properties of the Banach space. Quantitative results will be obtained in the finite-dimensional setting. A Banach space is a collection of objects called vectors which can be added together or multiplied by numbers to form other vectors. There is a concept of distance between vectors which is analogous to the familiar notion of distance between the points in the three-dimensional world which we inhabit. Mathematicians have found that Banach spaces provide the correct framework in which to formulate major areas of mathematics such as Functional Analysis and Partial Differential Equations. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, and finance. There are many different Banach spaces which can be distinguished from each other by geometrical properties such as smoothness and convexity. An individual vector belonging to a Banach space is usually identified by an infinite string of numbers called coefficients. An important problem in data compression is to find a method to select the most significant coefficients so that the resulting finite string vector is a short distance from the original vector. The project will concentrate on this and other problems involving geometrical properties of Banach spaces.