Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors
This work addresses the practical need for efficient sensing in signal processing by using structured matrices, though it is incremental as it extends existing guarantees to a more computationally efficient setting.
The paper tackles the problem of 1-bit compressive sensing with structured matrices, which are faster to compute than standard Gaussian matrices, and shows that a correlation-based algorithm achieves recovery guarantees comparable to those with i.i.d. Gaussian matrices, as supported by numerical experiments on image datasets.
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.