Sublinear Time, Measurement-Optimal, Sparse Recovery For All

TL;DR

This paper introduces a new sparse recovery algorithm that achieves optimal measurement complexity and sublinear runtime, significantly improving efficiency for large-scale signals in the all-in-one measurement model.

Contribution

It presents the first sublinear-time algorithm that uses optimal measurements for sparse recovery in the all signals model.

Findings

Uses O(k log(N/k)) measurements, optimal up to constants

Runs in o(N) time for k=o(N)

Requires a data structure with O(N) space and preprocessing

Abstract

An approximate sparse recovery system in ell_1 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 x_k denotes the optimal k-term approximation to x, the system approximates x by hat_x = D(Phi.x), which must satisfy ||hat_x - x||_1 <= (1+epsilon)||x - x_k||_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).