Sublinear time, measurement-optimal, sparse recovery for all

An approximate sparse recovery system in ℓ₁ norm makes a small number of measurements of a noisy vector with at most k large entries and recovers those heavy hitters approximately. Formally, it consists of parameters N, k, ε, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, where x_k denotes the optimal k-term approximation to x, the system approximates x by x̂ = D(Φx), which must satisfy ∥x̂ - x∥₁ ≤ (1 + ε) ∥x - x _k∥₁. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, V. We consider the "forall" model, in which a single matrix Φ, possibly "constructed" non-explicitly using the probabilistic method, is used for all signals x. Many previous papers have provided algorithms for this problem. But all such algorithms that use the optimal number m = O(k log(N/k)) of measurements require superlinear time Ω(N log(N/k)). In this paper, we give the first algorithm for this problem that uses the optimum number of measurements (up to constant factors) and runs in sublinear time o(N) when k is sufficiently less than N. Specifically, for any positive integer ℓ, our approach uses time O(ℓ⁵ ε^-3 k(N/k) ^1/ℓ) and uses m = O(ℓ⁸ ε^-3 k log(N/k)) measurements, with access to a data structure requiring space and preprocessing time O(ℓN k^0.2/ε).