# Sharp Guarantees for Solving Random Equations with One-Bit Information

## TL;DR

This paper provides sharp theoretical guarantees for convex optimization methods in recovering signals from one-bit measurements in high-dimensional settings, applicable to various loss functions and validated by simulations.

## Contribution

It offers a unified analysis predicting the performance of a broad class of convex estimators for one-bit compressed sensing, including new bounds for several loss functions.

## Key findings

- Performance predictions match simulations in high dimensions
- Bounds apply to multiple convex loss functions
- Results hold for Gaussian measurement vectors

## Abstract

We study the performance of a wide class of convex optimization-based estimators for recovering a signal from corrupted one-bit measurements in high-dimensions. Our general result predicts sharply the performance of such estimators in the linear asymptotic regime when the measurement vectors have entries IID Gaussian. This includes, as a special case, the previously studied least-squares estimator and various novel results for other popular estimators such as least-absolute deviations, hinge-loss and logistic-loss. Importantly, we exploit the fact that our analysis holds for generic convex loss functions to prove a bound on the best achievable performance across the entire class of estimators. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.