One-bit compressed sensing (1bCS) is an extreme-quantized signal acquisition method that has been widely studied in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign of the measurement). Assuming the original signal vector to be sparse, existing results either aim to find the support of the vector, or approximate the signal within an $\epsilon$-ball. The focus of this paper is support recovery, which often also computationally facilitates approximate signal recovery. A universal measurement matrix for 1bCS refers to one set of measurements that work for all sparse signals. With universality, it is known that $\tilde{\Theta}(k^2)$ 1bCS measurements are necessary and sufficient for support recovery (where $k$ denotes the sparsity). In this work, we show that it is possible to universally recover the support with a small number of false positives with $\tilde{O}(k^{3/2})$ measurements. If the dynamic range of the signal vector is known, then with a different technique, this result can be improved to only $\tilde{O}(k)$ measurements. Further results on support recovery are also provided.

中文翻译：

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支持通用一位压缩感知中的恢复

一位压缩感知（1bCS）是一种极端量化的信号采集方法，在过去的十年中得到了广泛的研究。在 1bCS 中，高维信号的线性样本被量化为每个样本仅一位（测量的符号）。假设原始信号向量是稀疏的，现有结果要么旨在找到向量的支持，要么在 $\epsilon$-ball 内近似信号。本文的重点是支持恢复，这通常在计算上也有助于近似信号恢复。1bCS 的通用测量矩阵是指一组适用于所有稀疏信号的测量。具有普遍性，众所周知，$\tilde{\Theta}(k^2)$ 1bCS 测量对于支持恢复是必要且充分的（其中 $k$ 表示稀疏性）。在这项工作中，我们表明，可以使用 $\tilde{O}(k^{3/2})$ 测量值以少量误报普遍恢复支持。如果信号向量的动态范围已知，那么使用不同的技术，可以将这个结果改进为仅 $\tilde{O}(k)$ 测量。还提供了有关支持恢复的进一步结果。