Achievable Angles Between Two Compressed Sparse Vectors Under RIP-Induced Norm/Distance Constraints

Abstract

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 an important role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors with measured angle(u, v) = theta and (ii) the sensing matrix satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between u and 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 RIPinduced norm/distance constraints on u and 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, closedform 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.