713 research outputs found

    Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property

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    Compressed Sensing aims to capture attributes of kk-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the \m\times \n measurement matrix \A is required to act as a near isometry on the set of all kk-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate \m \times \n matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix \A has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of kk-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in \n, and only quadratic in \m; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

    Deterministic Sampling of Sparse Trigonometric Polynomials

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    One can recover sparse multivariate trigonometric polynomials from few randomly taken samples with high probability (as shown by Kunis and Rauhut). We give a deterministic sampling of multivariate trigonometric polynomials inspired by Weil's exponential sum. Our sampling can produce a deterministic matrix satisfying the statistical restricted isometry property, and also nearly optimal Grassmannian frames. We show that one can exactly reconstruct every MM-sparse multivariate trigonometric polynomial with fixed degree and of length DD from the determinant sampling XX, using the orthogonal matching pursuit, and # X is a prime number greater than (MlogD)2(M\log D)^2. This result is almost optimal within the (logD)2(\log D)^2 factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.Comment: 9 page

    Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

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    In this paper, we study the problem of compressed sensing using binary measurement matrices and 1\ell_1-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions % sparsity bounds for robust sparse recovery obtained using the RNSP are better by a factor of (33)/22.6(3 \sqrt{3})/2 \approx 2.6 compared to the sufficient conditions obtained using the restricted isometry property (RIP). Next we derive universal \textit{lower} bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes and Euler squares, which have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle, randomly generated Gaussian measurement matrices are "order-optimal". So we compare the phase transition behavior of the basis pursuit formulation using binary array codes and Gaussian matrices and show that (i) there is essentially no difference between the phase transition boundaries in the two cases and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table

    The road to deterministic matrices with the restricted isometry property

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    The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page
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