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

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 1) u and v are two sparse vectors with
$\measuredangle({\bf u},{\bf v})=\theta$
and 2) the sensing matrix
${\mmb\Phi}$
satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between
${\mmb\Phi}{\bf u}$
and
${\mmb\Phi}{\bf v}$
. 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
${\mmb\Phi}{\bf u}$
and
${\mmb\Phi}{\bf v}$
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.