Improved bounds for universal one-bit compressive sensing

Jayadev Acharya, Arnab Bhattacharyya, Pritish Kamath

Abstract: Unlike compressive sensing where the measurement outputs are assumed to be real-valued and have infinite precision, in one-bit compressive sensing, measurements are quantized to one bit, their signs. In this work, our contributions are as follows: 1. We show how to recover the support of sparse high-dimensional vectors in the 1-bit compressive sensing framework with an asymptotically near-optimal number of measurements. We do this by showing an equivalence between the task of support recovery using 1-bit compressive sensing and a well-studied combinatorial object known as Union Free Families. 2. We also improve the bounds on the number of measurements for approximately recovering vectors from 1-bit compressive sensing measurements. All our results are about universal measurements, namely the measurement schemes that work simultaneously for all sparse vectors. Our improved bounds naturally lead the way to suggest several interesting open problems.