For-all sparse recovery in near-optimal time

An approximate sparse recovery system in ℓ₁ norm consists of parameters k, ε, N, an m-by-N measurement Φ, and a recovery algorithm, R. Given a vector, x, the system approximates x by x̂ = R(Φx), which must satisfy ∥ x̂-x∥1 ≤ (1 + ε)∥x - x _k∥1. We consider the "for all" model, in which a single matrix Φ is used for all signals x. The best existing sublinear algorithm by Porat and Strauss (SODA'12) uses O(ε^-3 klog(N/k)) measurements and runs in time O(k^1-α N^α) for any constant α>0. In this paper, we improve the number of measurements to O(ε^-2 k log(N/k)), matching the best existing upper bound (attained by super-linear algorithms), and the runtime to O(k^1+β poly(logN,1/ε)), with a modest restriction that k ≤ N ^1-α and ε ≤ (logk/logN)^γ, for any constants α, β,γ > 0. With no restrictions on ε, we have an approximation recovery system with m = O(k/εlog(N/k)((logN/logk) ^γ +1/ε)) measurements. The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages.