Support Recovery in One-bit Compressed Sensing with Near-Optimal Measurements and Sublinear Time
Xiaxin Li, Arya Mazumdar
TL;DR
This work addresses the problem of recovering the support of a $k$-sparse signal $x\in\mathbb{R}^n$ from one-bit measurements $y=\text{sign}(Ax)$, aiming to beat the standard $\Omega(n)$ decoding time. It introduces EDOCS, a framework yielding sublinear-time recovery via two universal schemes—EDOCS-AA for universal $\varepsilon$-approximate recovery with $m=O(k\varepsilon^{-1}\log(n/k)\log n)$ and $D=O(\varepsilon^{-1}m)$, and EDOCS-AE for universal exact recovery with $m=O(k^2\log(n/k)\log n)$ and $D=O(km)$—plus a probabilistic exact sublinear scheme with $m=O\left(k\frac{\log k}{\log\log k}\log n\right)$ and $D=O(m)$ that improves prior results. The universal schemes rely on combinatorial matrices (list union-free and distinguishable) and block decoding inspired by SAFFRON, while the probabilistic scheme adapts fast binary splitting from group testing and maps it to 1bCS using AsTIM-based constructions to control false positives. Collectively, the results achieve substantially reduced decoding time relative to previous $O(n)$ approaches, at the expense of modest measurement overhead in certain regimes, and open avenues for robust, noisy, or universal superset variants. These advances offer practical gains for rapid support recovery in highly quantized sensing applications and lay groundwork for further sublinear, robust 1bCS solutions.
Abstract
The problem of support recovery in one-bit compressed sensing (1bCS) aim to recover the support of a signal $x\in \mathbb{R}^n$, denoted as supp$(x)$, from the observation $y=\text{sign}(Ax)$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix and $|\text{supp}(x)|\leq k, k \ll n$. Under this setting, most preexisting works have a recovery runtime $Ω(n)$. In this paper, we propose two schemes that have sublinear $o(n)$ runtime. (1): For the universal exact support recovery, a scheme of $m=O(k^2\log(n/k)\log n)$ measurements and runtime $D=O(km)$. For the universal $ε$-approximate support recovery, the same scheme with $m=O(kε^{-1}\log(n/k)\log n)$ and runtime $D=O(ε^{-1}m)$, improving the runtime significantly with an extra $O(\log n)$ factor in the number of measurements compared to the current optimal (Matsumoto et al., 2023). (2): For the probabilistic exact support recovery in the sublinear regime, a scheme of $m:=O(k\frac{\log k}{\log\log k}\log n)$ measurements and runtime $O(m)$, with vanishing error probability, improving the recent result of Yang et al., 2025.