# Improved Bounds for Universal One-Bit Compressive Sensing

## TL;DR

This paper presents new bounds and methods for support recovery and approximate vector recovery in one-bit compressive sensing, achieving near-optimal measurement efficiency with universal schemes.

## Contribution

It introduces improved measurement bounds for support and approximate recovery in one-bit compressive sensing, establishing universality and optimality via combinatorial equivalences.

## Key findings

- Support recovery with near-optimal measurements
- Universal measurement schemes for all sparse vectors
- Improved bounds on measurement numbers for approximate recovery

## 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, we show how to recover the support of sparse high-dimensional vectors in the one-bit compressive sensing framework with an asymptotically near-optimal number of measurements. We also improve the bounds on the number of measurements for approximately recovering vectors from one-bit compressive sensing measurements. Our results are universal, namely the same measurement scheme works simultaneously for all sparse vectors. Our proof of optimality for support recovery is obtained 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.