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The goal of standard 1-bit compressive sensing is to accurately recover anunknown sparse vector from binary-valued measurements, each indicating the signof a linear function of the vector. Motivated by recent advances in compressivesensing with generative models, where a generative modeling assumption replacesthe usual sparsity assumption, we study the problem of 1-bit compressivesensing with generative models. We first consider noiseless 1-bit measurements,and provide sample complexity bounds for approximate recovery underi.i.d.~Gaussian measurements and a Lipschitz continuous generative prior, aswell as a near-matching algorithm-independent lower bound. Moreover, wedemonstrate that the Binary $\epsilon$-Stable Embedding property, whichcharacterizes the robustness of the reconstruction to measurement errors andnoise, also holds for 1-bit compressive sensing with Lipschitz continuousgenerative models with sufficiently many Gaussian measurements. In addition, weapply our results to neural network generative models, and provide aproof-of-concept numerical experiment demonstrating significant improvementsover sparsity-based approaches.

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