For-All Sparse Recovery in Near-Optimal Time

Abstract

An approximate sparse recovery system in ℓ₁ norm consists of parameters k, ε, N, an m-by-N measurement Φ, and a recovery algorithm, $\mathcal{R}$. Given a vector, x, the system approximates x by $\widehat{\mathbf{x}} = \mathcal{R}(\Phi\mathbf{x})$, which must satisfy $\|\widehat{\mathbf{x}}-\mathbf{x}\|_1 \leq (1+\epsilon)\|\mathbf{x}-\mathbf{x}_k\|_1$. 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 + βpoly(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.

Omitted details and proofs can be found at arXiv:1402.1726 [cs.DS].