Practical near-optimal sparse recovery in the L1 norm

We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector x isin Rn from its lower-dimensional sketch Ax isin Rm. Specifically, we focus on the sparse recovery problem in the l1 norm: for a parameter k, given the sketch Ax, compute an approximation xcirc of x such that the l1 approximation error parx - xcircpar1 is close to minx' parx - x'par1, where x' ranges over all vectors with at most k terms. The sparse recovery problem has been subject to extensive research over the last few years. Many solutions to this problem have been discovered, achieving different trade-offs between various attributes, such as the sketch length, encoding and recovery times.