One-bit compressed sensing with partial Gaussian circulant matrices

TL;DR

This paper demonstrates that partial Gaussian circulant matrices enable efficient one-bit compressed sensing for sparse signals, requiring a number of measurements that scales with sparsity, accuracy, and signal dimension.

Contribution

It proves that partial Gaussian circulant matrices satisfy an $ ext{L}_1/ ext{L}_2$ RIP, enabling stable and accurate one-bit compressed sensing with fewer measurements.

Findings

Measurements scale as $ ext{delta}^{-4} s ext{log}(N/s ext{delta})$ for accurate recovery.

Partial Gaussian circulant matrices satisfy the $ ext{L}_1/ ext{L}_2$ RIP property.

Stable recovery is possible even with approximate sparsity.

Abstract

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\sim \delta^{-4} s\log(N/s\delta)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.