A sublinear algorithm for sparse reconstruction with ℓ2/ℓ2 recovery guarantees

Compressed sensing aims to capture attributes of a sparse signal using very few measurements. Candes and Tao showed that sparse reconstruction is possible if the sensing matrix acts as a near isometry on all k-sparse signals. This property holds with overwhelming probability if the entries of the matrix are generated by an iid Gaussian or Bernoulli process. There has been significant recent interest in an alternative signal processing framework; exploiting deterministic sensing matrices that with overwhelming probability act as a near isometry on k-sparse vectors with uniformly random support, a geometric condition that is called the Statistical Restricted Isometry Property or StRIP. This paper considers a family of deterministic sensing matrices satisfying the StRIP that are based on Delsarte-Goethals Codes codes (binary chirps) and a k-sparse reconstruction algorithm with sublinear complexity. In the presence of stochastic noise in the data domain, this paper derives bounds on the l₂ accuracy of approximation in terms of the l₂ norm of the measurement noise and the accuracy of the best k-sparse approximation, also measured in the l₂ norm. This type of l₂/l₂ bound is tighter than the standard l₂/l₁ or l₁/l₁ bounds.