## 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) = θ and (ii) the sensing matrix Φ satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Φu and Φv. 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 Φu and Φv 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, closed-form 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.

Original language | English |
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Pages | 523-528 |

Number of pages | 6 |

DOIs | |

State | Published - 2014 |

Event | 2014 International Conference on Computing, Networking and Communications, ICNC 2014 - Honolulu, HI, United States Duration: 3 Feb 2014 → 6 Feb 2014 |

### Conference

Conference | 2014 International Conference on Computing, Networking and Communications, ICNC 2014 |
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Country/Territory | United States |

City | Honolulu, HI |

Period | 3/02/14 → 6/02/14 |