  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.
