The goal of standard 1-bit compressive sensing is to accurately recover an
unknown sparse vector from binary-valued measurements, each indicating the sign
of a linear function of the vector. Motivated by recent advances in compressive
sensing with generative models, where a generative modeling assumption replaces
the usual sparsity assumption, we study the problem of 1-bit compressive
sensing with generative models. We first consider noiseless 1-bit measurements,
and provide sample complexity bounds for approximate recovery under
i.i.d.~Gaussian measurements and a Lipschitz continuous generative prior, as
well as a near-matching algorithm-independent lower bound. Moreover, we
demonstrate that the Binary $\epsilon$-Stable Embedding property, which
characterizes the robustness of the reconstruction to measurement errors and
noise, also holds for 1-bit compressive sensing with Lipschitz continuous
generative models with sufficiently many Gaussian measurements. In addition, we
apply our results to neural network generative models, and provide a
proof-of-concept numerical experiment demonstrating significant improvements
over sparsity-based approaches.

该研究利用生成模型替代稀疏性假设，研究带生成模型的 1 位压缩感知问题。在此基础上结合高斯测量和具有 Lipschitz 连续生成先验的近似恢复，应用于神经网络生成模型，并与基于稀疏性的方法进行了比较，证实了其有效性。

具有生成 先验的 1 位压缩感知和二进制稳定嵌入的样本复杂度界限

Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable  Embeddings with Generative Priors

The Compressive Sensing (CS) framework aims to ease the burden on
analog-to-digital converters (ADCs) by reducing the sampling rate required to
acquire and stably recover sparse signals. Practical ADCs not only sample but
also quantize each measurement to a finite number of bits; moreover, there is
an inverse relationship between the achievable sampling rate and the bit depth.
In this paper, we investigate an alternative CS approach that shifts the
emphasis from the sampling rate to the number of bits per measurement. In
particular, we explore the extreme case of 1-bit CS measurements, which capture
just their sign. Our results come in two flavors. First, we consider ideal
reconstruction from noiseless 1-bit measurements and provide a lower bound on
the best achievable reconstruction error. We also demonstrate that i.i.d.
random Gaussian matrices describe measurement mappings achieving, with
overwhelming probability, nearly optimal error decay. Next, we consider
reconstruction robustness to measurement errors and noise and introduce the
Binary $\epsilon$-Stable Embedding (B$\epsilon$SE) property, which
characterizes the robustness measurement process to sign changes. We show the
same class of matrices that provide almost optimal noiseless performance also
enable such a robust mapping. On the practical side, we introduce the Binary
Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from
1-bit measurements that offers state-of-the-art performance.

该论文研究了利用压缩感知技术来减少模数转换器的采样率，着重探讨了只捕捉信号符号的 1 比特压缩感知技术，提出了能在噪声较小及容易受干扰的情况下实现逆向恢复的二进制 ε- 稳定嵌入特性，并引入了适用于 1 比特采样的算法 —— 二进制迭代硬阈值算法，提高了采样性能。