基于部分高斯循环矩阵和生成先验的鲁棒1比特压缩感知
Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors
摘要：
在1比特压缩感知中，每个测量被量化为单个比特，即未知向量的线性函数的符号，目标是准确地恢复向量。虽然最流行的是假设标准高斯感测矩阵用于1比特压缩感测，但是使用结构化感测矩阵（诸如部分高斯循环矩阵）由于其更快的矩阵运算而具有显著的实际重要性。在本文中，我们提供了一个基于相关性的鲁棒1位压缩感知的优化算法与随机签署的部分高斯循环矩阵和生成模型的恢复保证。在适当的假设下，我们匹配了以前只知道对i.i.d.~高斯矩阵，需要更多的计算。我们利用一个实用的迭代算法，并进行了数值实验上的图像数据集，以证实我们的理论结果。
英文摘要：
In 1-bit compressive sensing, each measurement is quantized to a single bit, namely the sign of a linear function of an unknown vector, and the goal is to accurately recover the vector. While it is most popular to assume a standard Gaussian sensing matrix for 1-bit compressive sensing, using structured sensing matrices such as partial Gaussian circulant matrices is of significant practical importance due to their faster matrix operations. In this paper, we provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing with randomly signed partial Gaussian circulant matrices and generative models. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d.~Gaussian matrices that require significantly more computation. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our theoretical results.
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