Deterministic Compressive Sensing with Groups of Random Variables

Compressed Sensing aims to capture attributes of a sparse signal using very few measurements. Candès and Tao showed that random matrices satisfy the Restricted Isometry Property or RIP property with high probability. They showed that this condition is sufficient for sparse reconstruction and that random matrices, where the entries are generated by an iid Gaussian or Bernoulli process, satisfy the RIP with high probability. This approach treats all k-sparse signals as equally likely, in contrast to mainstream signal processing where the filtering is deterministic, and the signal is described probabilistically. This paper provides weak conditions that are sufficient to show that a deterministic sensing matrix satisfies the Statistical Restricted Isometry Property (StRIP), and to guarantee the uniqueness of k-sparse representations. The columns of the sensing matrix are required to form a group under pointwise multiplication, and it is this property that provides the concentration inequalities of McDiarmid and Bernstein with the leverage necessary to guarantee uniqueness of sparse representations. We provide a large class of deterministic sensing matrices for which Basis Pursuit is able to recover sparse Steinhaus signals. The new framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and Extended BCH codes. In these cases it provides theoretical guarantees on the performance of nonlinear reconstruction algorithms with complexity that is only quadratic in the dimension of the measurement domain.