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How Close Two Sparse Vectors in the Compressive Domain Can Be: Compressive Sensing Meets Plane Geometry

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How Close Two Sparse Vectors in the Compressive Domain Can Be: Compressive Sensing Meets Plane Geometry

  • LecturerProf. Jwo-Yuh Wu (Department of Electrical and Computer Engineering, and the Institute of Communications Engineering, National Chiao Tung University)
    Host: Chun-Shien Lu
  • Time2014-11-19 (Wed.) 14:00 ~ 16:00
  • LocationAuditorium 106 at new IIS Building
Abstract

The angle between two compressed sparse vectors subject to the norm/distanceconstraints imposed by the restricted isometry property (RIP) of the sensing matrix plays animportantrole in the studies ofmany compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors with (lambda) under bar (u, v) = theta and (ii) the sensing matrix Phi satisfies RIP, our purpose is to analytically characterize 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 involved algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the maximaland 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.Applications of our results in CS are also discussed.

BIO

Jwo-Yuh Wu received the B. S. degree in 1996, the M. S. degree in 1998, and the Ph. D. degree in 2002, all in Electrical and Control Engineering, National Chiao Tung University, Taiwan. During 2003 and 2007, he was a post doctor research fellow in the Department of Communications Engineering, National Chiao Tung University, Taiwan. Starting from 2008, he was a faculty member in the Department of Electrical and Computer Engineering, and the Institute of Communications Engineering, National Chiao Tung University, Taiwan, where he has been an associate professor since 2011. His general research interests are in signal processing, wireless communications, control systems, linear algebra, and applied functional analysis. Dr. Wu received the Ta-You Wu Memorial Award from Ministry of Science and Technology of Taiwan in 2014,the 12thY. Z. Hsu Scientific Paper Award (Communication Technology Category) in 2014, and 2013 Best Paper Award for Young Scholars by IEEE Information Theory Society Taipei Chapter and IEEE Communications Society Taipei/Tainan Chapter.