Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

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 partial Gaussian circulant matrices (with random column sign flips) under a generative prior, where the signal to estimate is assumed to belong to the range of a Lipschitz continuous generative model with bounded inputs. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d.~Gaussian matrices requiring significantly more computation.