One-bit compressed sensing by linear programming

TL;DR

This paper introduces a computationally feasible linear programming method for one-bit compressed sensing, enabling accurate recovery of sparse vectors from minimal sign measurements with high probability.

Contribution

It provides the first efficient and nearly optimal linear programming approach for one-bit compressed sensing, extending to approximately sparse vectors and ensuring universality.

Findings

Recovery from O(s log^2(n/s)) measurements

Method works for approximately sparse vectors

Universal measurement scheme with high probability

Abstract

We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.