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

TL;DR

This paper introduces a robust method for 1-bit compressive sensing using structured partial Gaussian circulant matrices and generative priors, achieving reliable recovery with faster computations.

Contribution

It provides theoretical recovery guarantees for a correlation-based algorithm with structured matrices, extending results previously limited to Gaussian matrices.

Findings

The proposed method achieves accurate recovery in experiments on image datasets.

Theoretical guarantees match those for i.i.d. Gaussian matrices under certain conditions.

Structured matrices enable faster computations without sacrificing recovery performance.

Abstract

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.