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.

[1]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[2]  Sergio Verdu,et al.  Multiuser Detection , 1998 .

[3]  Steven Kay,et al.  Fundamentals Of Statistical Signal Processing , 2001 .

[4]  Bernhard Schölkopf,et al.  Sampling Techniques for Kernel Methods , 2001, NIPS.

[5]  Avner Magen,et al.  Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications , 2002, RANDOM.

[6]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[7]  Robert D. Nowak,et al.  Signal Reconstruction From Noisy Random Projections , 2006, IEEE Transactions on Information Theory.

[8]  Richard G. Baraniuk,et al.  Detection and estimation with compressive measurements , 2006 .

[9]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[10]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[11]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[12]  Robert Nowak,et al.  Active learning versus compressive sampling , 2006, SPIE Defense + Commercial Sensing.

[13]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[14]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .