Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing

2026-10-08

AI Summary

This study addresses the lack of theoretical guarantees and efficient algorithms for signal recovery sample complexity in noisy one-bit compressed sensing under prior distributions. By characterizing distributional complexity via approximate covering numbers and incorporating the Wasserstein distance metric, this work establishes the first upper and lower bounds for posterior sampling that are robust to prior mismatch. Furthermore, it introduces a plug-and-play algorithm leveraging diffusion priors to approximate ideal posterior sampling. The proposed framework reveals the one-bit separation gap factor and constructs a near-optimal sample complexity theory with nearly matching upper and lower bounds. Empirical evaluations on the FFHQ and ImageNet datasets validate the practical effectiveness of the proposed method.

Abstract

We study the sample complexity of noisy one-bit compressed sensing for signals drawn from a prior distribution. By characterizing the effective distributional complexity of the prior via its approximate covering number, we prove that posterior sampling achieves accurate recovery with high probability when the number of measurements scales with the logarithm of the approximate covering number, up to a one-bit separation gap factor. This upper bound is robust to learned prior mismatch. Specifically, we show that posterior sampling with an approximate prior remains reliable, provided that the learned prior distribution is sufficiently close to the true signal distribution in Wasserstein distance. In addition, we establish a sample complexity lower bound for any reliable method of noisy one-bit compressed sensing, showing that our upper bound is nearly matched in its main prior dependent term. To approximate the ideal posterior sampling process for real world scenarios, we instantiate posterior sampling through a plug-and-play algorithm with diffusion priors. Experiments on the FFHQ and ImageNet datasets demonstrate the effectiveness of our proposed approach.

Problem

• One-Bit Compressed Sensing
• Sample Complexity
• Posterior Sampling
• Prior Mismatch
• Recovery Guarantees

Innovation

• One-Bit Compressed Sensing
• Posterior Sampling
• Sample Complexity
• Diffusion Priors
• Wasserstein Distance