One-Bit Compressed Sensing via One-Shot Hard Thresholding
It addresses the problem of estimating sparse signals from binary measurements for applications like signal processing, offering incremental improvements in analysis and computational efficiency.
The paper tackles 1-bit compressed sensing by proposing a non-convex sparsity-constrained program and a simple algorithm that produces an accurate approximation to the normalized sparse signal under the ℓ₂-metric with high probability, achieving near-optimal error rates and demonstrating dramatic efficiency via one-step hard thresholding.
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.