One-bit quantization with time-varying sampling thresholds (also known as random dithering) has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to-digital converters (ADCs) is their superior sampling rates as compared to their conventional multi-bit counterparts. This characteristic endows one-bit signal processing frameworks with what one may refer to as sample abundance. We show that sample abundance plays a pivotal role in many signal recovery and optimization problems that are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints. Of particular interest to our work are low-rank matrix recovery and compressed sensing applications that take advantage of one-bit quantization. We demonstrate that the sample abundance paradigm allows for the transformation of such problems to merely linear feasibility problems by forming large-scale overdetermined linear systems -- thus removing the need for handling costly optimization constraints and objectives. To make the proposed computational cost savings achievable, we offer enhanced randomized Kaczmarz algorithms to solve these highly overdetermined feasibility problems and provide theoretical guarantees in terms of their convergence, sample size requirements, and overall performance. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

翻译：时变采样阈值（又称随机抖动）的单比特量化因功耗低、实现成本低，近年来在统计信号处理应用中展现出巨大的应用潜力。除上述优势外，单比特模数转换器相比传统多比特模数转换器具有更高的采样率。这种特性赋予了单比特信号处理框架所谓的“样本丰度”。我们证明，在众多可表述为（可能非凸的）二次规划与线性可行性约束的信号恢复与优化问题中，样本丰度发挥着关键作用。我们特别关注利用单比特量化的低秩矩阵恢复与压缩感知应用，并论证了样本丰度范式可将此类问题转化为纯粹的线性可行性问题（通过构建大规模超定线性方程组），从而消除处理昂贵优化约束与目标的需求。为实现所提出的计算成本节约，我们提出了增强型随机卡茨马尔兹算法来解决这些高度超定的可行性问题，并提供了收敛性、样本量需求及整体性能的理论保证。最后通过数值实验结果展示了所提方法的有效性。