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 ∈ R
ⁿ
from its lower-dimensional sketch Ax ∈ R
^m
. Specifically, we focus on the sparse recovery problem in the l
1
norm: for a parameter k, given the sketch Ax, compute an approximation x̂ of x such that the l
1
approximation error ‖x − x̂‖
1
is close to min
x′
‖x − x′‖
1
, 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.