Summary based structures with improved sublinear recovery for compressed sensing

We introduce a new class of measurement matrices for compressed sensing, using low order summaries over binary sequences of a given length. We prove recovery guarantees for three reconstruction algorithms using the proposed measurements, including l₁ minimization and two combinatorial methods. In particular, one of the algorithms recovers k-sparse vectors of length N in sublinear time poly(k log N), and requires at most O(k log N log log N) measurements. The empirical oversampling constant of the algorithm is significantly better than existing sublinear recovery algorithms such as Chaining Pursuit and Sudocodes. In particular, for 10³ ≤ N ≤ 10¹² and k = 100, the oversampling factor is between 5 to 25. We provide preliminary insight into how the proposed constructions, and the fast recovery scheme can be used in a number of practical applications such as market basket analysis, and real time compressed sensing implementation.