For-All Sparse Recovery in Near-Optimal Time

Abstract

AbstractAn 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.KeywordsNetwork CodeMeasurement MatrixRecovery AlgorithmExpander GraphWeak SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.