Construction of a Large Class of Deterministic Sensing Matrices That Satisfy a Statistical Isometry Property
IEEE Journal on Selected Topics in Signal Processing
Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the × measurement matrix is required to act as a near isometry on the set of all k-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate × matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix has this property, crucial
... or the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of k-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in , and only quadratic in ; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.