One-Bit Compressed Sensing via One-Shot Hard Thresholding

TL;DR

This paper introduces a simple, efficient algorithm for 1-bit compressed sensing that accurately estimates sparse signals from binary measurements, with strong theoretical guarantees and practical validation.

Contribution

It presents a novel analysis framework for 1-bit compressed sensing and demonstrates that one-step hard thresholding achieves near-optimal error rates efficiently.

Findings

High-probability accurate signal approximation

Efficient one-step hard thresholding algorithm

Near-optimal error rates under standard conditions

Abstract

This paper concerns the problem of 1-bit compressed sensing, where the goal is to estimate a sparse signal from a few of its binary measurements. We study a non-convex sparsity-constrained program and present a novel and concise analysis that moves away from the widely used notion of Gaussian width. We show that with high probability a simple algorithm is guaranteed to produce an accurate approximation to the normalized signal of interest under the $\ell_2$-metric. On top of that, we establish an ensemble of new results that address norm estimation, support recovery, and model misspecification. On the computational side, it is shown that the non-convex program can be solved via one-step hard thresholding which is dramatically efficient in terms of time complexity and memory footprint. On the statistical side, it is shown that our estimator enjoys a near-optimal error rate under standard conditions. The theoretical results are substantiated by numerical experiments.