Construction of a Large Class of Deterministic Sensing Matrices That Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard compressed sensing paradigm, the N x C measurement matrix Phi 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 N x C matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix Phi has this property, crucial for 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. An essential element in our construction is that 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 N, as compared to the super-linear complexity in C of the Basis Pursuit or Matching Pursuit algorithms; 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.