Sublinear Time, Measurement-Optimal, Sparse Recovery For All
Abstract
An approximate sparse recovery system in ell1 norm formally consists of parameters N, k, epsilon an m-by-N measurement matrix, Phi, and a decoding algorithm, D. Given a vector, x, where xk denotes the optimal k-term approximation to x, the system approximates x by hatx = D(Phi.x), which must satisfy ||hatx - x||1 <= (1+epsilon)||x - xk||1. Among the goals in designing such systems are minimizing m and the runtime of D. We consider the "forall" model, in which a single matrix Phi is used for all signals x. All previous algorithms that use the optimal number m=O(k log(N/k)) of measurements require superlinear time Omega(N log(N/k)). In this paper, we give the first algorithm for this problem that uses the optimum number of measurements (up to a constant factor) and runs in sublinear time o(N) when k=o(N), assuming access to a data structure requiring space and preprocessing O(N).