Optimizing privacy and utility in one-bit compressive sensing with adaptive local differential privacy

One-bit compressive sensing (1-bit CS) offers significant hardware and computational cost advantages and has application prospects in low-resource overhead scenarios. However, achieving accurate signal reconstruction while rigorously protecting data privacy in 1-bit CS systems, which are highly sensitive to noise, remains a substantial challenge, as traditional Differential Privacy (DP) methods often struggle to balance this trade-off. To address this critical issue, this paper proposes a comprehensive framework that synergistically integrates an adaptive DP method with Bayesian inference. We propose a novel adaptive DP method that injects noise based on the characteristics of the measurements and adjusts the noise level according to the measurement matrix properties and data statistics. This is complemented by a new Bayesian reconstruction algorithm, specifically designed to effectively handle the heteroscedastic noise introduced by our adaptive DP method, thereby significantly improving signal recovery accuracy. Theoretical analysis confirms that the proposed method satisfies $(\varepsilon ,\delta )$-DP, while the reconstruction algorithm is proven to converge linearly under Restricted Isometry Property (RIP) conditions and achieves favorable reconstruction error bounds. Extensive experimental results demonstrate that our framework attains superior reconstruction performance under various privacy budgets, signal sparsities, and measurement ratios, consistently outperforming existing methods in privacy-preserving 1-bit CS scenarios.