A Generalized Restricted Isometry Property

Compressive Sampling (CS) describes a method for reconstructing high-dimensional sparse signals from a small number of linear measurements. Fundamental to the success of CS is the existence of special measurement matrices which satisfy the so-called Restricted Isometry Property (RIP). In essence, a matrix satisfying RIP is such that the lengths of all sufficiently sparse vectors are approximately preserved under transformation by the matrix. In this paper we describe a natural consequence of this property – if a matrix satisfies RIP, then acute angles between sparse vectors are also approximately preserved. We formulate this property as a Generalized Restricted Isometry Property (GRIP) and describe one application in robust signal detection.