Compressive sensing using locality-preserving matrices

Compressive sensing is a method for acquiring high dimensional signals (e.g., images) using a small number of linear measurements. Consider an n-pixel image x ∈ Rn, where each pixel p has value xp. The image is acquired by computing the measurement vector Ax, where A is an m x n measurement matrix, for some m << n. The goal is to design the matrix A and the recovery algorithm which, given Ax, returns an approximation to x. It is known that m=O(k log(n/k)) measurements suffices to recover the k-sparse approximation of x. Unfortunately, this result uses matrices A that are random. Such matrices are difficult to implement in physical devices. In this paper we propose compressive sensing schemes that use matrices A that achieve the near-optimal bound of m=O(k log n), while being highly "local". We also show impossibility results for stronger notions of locality.