SHO-FA: Robust Compressive Sensing with Order-Optimal Complexity, Measurements, and Bits

Suppose x is any exactly k-sparse vector in Rn. We present a class of sparse matrices A, and a corresponding algorithm that we call short and fast1 (SHO-FA) that, with high probability over A, can reconstruct x from Ax. The SHO-FA algorithm is related to the invertible bloom lookup tables recently introduced by Goodrich et al., with two important distinctions- SHO-FA relies on linear measurements, and is robust to noise. The SHO-FA algorithm is the first to simultaneously have the following properties: 1) it requires only O(k) measurements; 2) the bit precision of each measurement and each arithmetic operation is O (log(n) + P) (here, 2^-P corresponds to the desired relative error in the reconstruction of x); 3) the computational complexity of decoding is O(k) arithmetic operations and that of encoding is O(n) arithmetic operations; and 4) if the reconstruction goal is simply to recover a single component of x instead of all of x, with significant probability over A, this can be done in constant time. All the above constants are independent of all problem parameters other than the desired probability of success. For a wide range of parameters, these properties are informationtheoretically order-optimal. In addition, our SHO-FA algorithm works over fairly general ensembles of sparse random matrices, and is robust to random noise and (random) approximate sparsity for a large range of k. In particular, suppose the measured vector equals A(x + z) + e, where z and e correspond to the source tail and measurement noise, respectively. Under reasonable statistical assumptions on z and e, our decoding algorithm reconstructs x with an estimation error of O(||z||2 + ||e||2). The SHO-FA algorithm works with high probability over A, z, and e, and still requires only O(k) steps and O(k) measurements over O(log(n))-bit numbers. This is in contrast to most existing algorithms that focus on the worst case z model, where it is known that Ω(k log(n/k)) measurements over O(log(n))-bit numbers are necessary. Our algorithm has good empirical performance, as validated by simulations.