Title: Achievable Angles Between Two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach
Authors: Chang, Ling-Hua
Wu, Jwo-Yuh
Undergraduate Honors Program of Electrical Engineering and Computer Science
Keywords: Compressive sensing (CS);plane geometry;restricted isometry constant (RIC);restricted isometry property (RIP)
Issue Date: 1-Apr-2013
Abstract: The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensingmatrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that 1) u and v are two sparse vectors with (lambda) under bar (u, v) = theta and 2) 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 2-D 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 formulas 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 in CS are also discussed.
URI: http://dx.doi.org/10.1109/TIT.2012.2234825
ISSN: 0018-9448
DOI: 10.1109/TIT.2012.2234825
Volume: 59
Issue: 4
Begin Page: 2059
End Page: 2081
Appears in Collections:Articles

Files in This Item:

  1. 000316407900007.pdf