# One-Bit Sensing of Low-Rank and Bisparse Matrices

## TL;DR

This paper investigates the limits of recovering low-rank and bisparse matrices from one-bit measurements, providing theoretical error decay rates and practical algorithms for improved recovery accuracy.

## Contribution

It introduces the concept of consistency width for error bounds and proposes a practical recovery algorithm with near-optimal decay rates for Gaussian sensing schemes.

## Key findings

- Theoretical error decay rate is established using consistency width.
- An idealized recovery method achieves fourth-root decay of optimal rate.
- A practical algorithm attains sixth-root decay, close to optimal.

## Abstract

This note studies the worst-case recovery error of low-rank and bisparse matrices as a function of the number of one-bit measurements used to acquire them. First, by way of the concept of consistency width, precise estimates are given on how fast the recovery error can in theory decay. Next, an idealized recovery method is proved to reach the fourth-root of the optimal decay rate for Gaussian sensing schemes. This idealized method being impractical, an implementable recovery algorithm is finally proposed in the context of factorized Gaussian sensing schemes. It is shown to provide a recovery error decaying as the sixth-root of the optimal rate.