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

• URL: http://arxiv.org/abs/2002.01697v3
• Date: Mon, 22 Jun 2020 04:57:09 GMT
• Title: Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors
• Authors: Zhaoqiang Liu, Selwyn Gomes, Avtansh Tiwari, Jonathan Scarlett
• Abstract summary: Motivated by advances in compressive sensing with generative models, 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. 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.
• Score: 52.06292503723978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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