1,675 research outputs found

    Achievable Angles Between two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach

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    The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensing matrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors separated by an angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Phi*u and Phi*v. Motivated by geometric interpretations of RIP and with the aid of the well-known law of cosines, we propose a plane geometry based formulation for the study of the considered problem. It is shown that all the RIP-induced norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a simple geometric diagram in the two-dimensional plane. This allows for a joint analysis of all the considered algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the maximal and minimal achievable angles are derived. Computer simulations confirm that the proposed solution is tighter than an existing algebraic-based estimate derived using the polarization identity. The obtained results are used to derive a tighter restricted isometry constant of structured sensing matrices of a certain kind, to wit, those in the form of a product of an orthogonal projection matrix and a random sensing matrix. Follow-up applications to three CS problems, namely, compressed-domain interference cancellation, RIP-based analysis of the orthogonal matching pursuit algorithm, and the study of democratic nature of random sensing matrices are investigated.Comment: submitted to IEEE Trans. Information Theor

    New Coherence and RIP Analysis for Weak Orthogonal Matching Pursuit

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    In this paper we define a new coherence index, named the global 2-coherence, of a given dictionary and study its relationship with the traditional mutual coherence and the restricted isometry constant. By exploring this relationship, we obtain more general results on sparse signal reconstruction using greedy algorithms in the compressive sensing (CS) framework. In particular, we obtain an improved bound over the best known results on the restricted isometry constant for successful recovery of sparse signals using orthogonal matching pursuit (OMP).Comment: arXiv admin note: substantial text overlap with arXiv:1307.194

    Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit

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    This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements -- L_1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of the L_1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity (in practice even logarithmic), and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle.Comment: This is the final version of the paper, including referee suggestion

    A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

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    This paper demonstrates that if the restricted isometry constant δK+1\delta_{K+1} of the measurement matrix AA satisfies δK+1<1K+1, \delta_{K+1} < \frac{1}{\sqrt{K}+1}, then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every KK--sparse signal x\mathbf{x} in KK iterations from A\x. By contrast, a matrix is also constructed with the restricted isometry constant δK+1=1K \delta_{K+1} = \frac{1}{\sqrt{K}} such that OMP can not recover some KK-sparse signal x\mathbf{x} in KK iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009
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