One-Bit Sensing of Low-Rank and Bisparse Matrices
Simon Foucart, Laurent Jacques
TL;DR
The paper addresses the problem of recovering matrices that are simultaneously low-rank and bisparse from one-bit linear measurements. It introduces the consistency width $CW^m$ to quantify worst-case recovery error under consistency and establishes near-tight bounds $c rs/m \le CW^m \le C rs/m \ln(nm/rs)$. It then presents an idealized Gaussian-sensing back-projection achieving a decay of order $\left(\frac{rs\ln(n/s)}{m}\right)^{1/4}$ and a practical multistep recovery under factorized Gaussian sensing achieving $\left(\frac{rs\ln(n/s)}{m}\right)^{1/6}$, highlighting a trade-off between computational feasibility and convergence rate. Together, these results characterize fundamental limits for one-bit recovery of structured matrices and provide implementable algorithms with provable error decay, relevant for quantized sensing and high-dimensional matrix recovery.
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.