One-bit compressed sensing (1bCS) addresses the recovery of sparse signals from highly quantized measurements, retaining only the sign of each linear measurement. In the support recovery setting, the goal is to identify $\text{supp}(x)$, the nonzero coordinates of an unknown signal $x \in \mathbb{R}^n$ from $y = \text{sign}(Ax)$, where $A \in \mathbb{R}^{m \times n}$ and $|\text{supp}(x)| \le k \ll n$. Existing methods minimize the number of measurements but often incur $Ω(n)$ decoding complexity, limiting large-scale applicability. We propose new 1bCS schemes that achieve sublinear decoding complexity while maintaining near-optimal measurement bounds. For universal support recovery, our framework provides: (i) exact recovery with $m = O(k^2 \log(n/k) \log n)$ measurements and decoding complexity $D=O(km)$, and (ii) $ε$-approximate recovery with $m = O(k ε^{-1} \log(n/k) \log n)$ and $D=O(ε^{-1} m)$. For probabilistic exact recovery, we design a scheme with $m = O\big(k \frac{\log k}{\log\log k} \log n\big)$ and $D=O(m)$, achieving vanishing error probability. Our approach leverages ideas from group testing to bridge classical sparse recovery techniques with modern algorithmic efficiency considerations, highlighting a new trade-off between compression efficiency and computational complexity.

翻译：[译] 一位压缩感知(1bCS)研究从高度量化的测量中恢复稀疏信号的问题，该问题仅保留每次线性测量的符号。在支持恢复场景中，目标是基于 $y = \text{sign}(Ax)$ 识别未知信号 $x \in \mathbb{R}^n$ 的支撑集 $\text{supp}(x)$（即非零坐标），其中 $A \in \mathbb{R}^{m \times n}$ 且 $|\text{supp}(x)| \le k \ll n$。现有方法虽能最小化测量数，但解码复杂度常达 $Ω(n)$，限制了大规模应用。我们提出新型1bCS方案，在维持近最优测量数界的同时实现亚线性解码复杂度。针对通用支持恢复，本文框架提供：(i) 精确恢复，测量数 $m = O(k^2 \log(n/k) \log n)$ 且解码复杂度 $D=O(km)$；(ii) $ε$-近似恢复，测量数 $m = O(k ε^{-1} \log(n/k) \log n)$ 且 $D=O(ε^{-1} m)$。针对概率精确恢复，我们设计了一种方案，其测量数 $m = O\big(k \frac{\log k}{\log\log k} \log n\big)$ 且 $D=O(m)$，可实现渐近消失的误差概率。本文方法借助群体测试思想将经典稀疏恢复技术与现代算法效率需求相衔接，揭示了压缩效率与计算复杂度之间的新权衡关系。