One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex lp (0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of lp effectively s-sparse signals which contains genuinely s-sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of lp effectively s-sparse signals with error order Ο((s/mlog(mn/s)){(2-p)/(2+p)}). We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any lp effectively s-sparse bounded signal x can be estimated using augmented lp minimization model and empirical distribution function method respectively. Especially, to recover lp effectively s-sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm.

中文翻译：

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通过 Ι _p (0 < p < 1)-最小化方法的一位压缩传感

一位压缩感知旨在从仅捕获其符号的极其量化的线性测量中恢复未知的稀疏信号。在本文中，我们为一位压缩感知问题提出了一个非凸 lp (0 < p < 1) 最小化模型，并定义了一组 lp 有效 s 稀疏信号，其中包含真正的 s 稀疏信号。利用覆盖数的性质，我们表明我们的方法可以有效地恢复 lp 的方向，错误顺序为 Ο((s/mlog(mn/s)){(2-p)/(2+p)} ）。我们还使用阈值一位测量来估计信号的幅度，并证明可以分别使用增强的 lp 最小化模型和经验分布函数方法来估计任何 lp 有效的 s 稀疏有界信号 x。特别是，为了在实践中有效地恢复 lp s 稀疏信号，我们引入了一种自适应二进制迭代阈值算法，该算法可以在不知道基础信号稀疏性的情况下使用。对合成数据集和真实数据集进行了数值实验，以证明我们算法的优越性。