We explore the impact of coarse quantization on matrix completion in the extreme scenario of dithered one-bit sensing, where the matrix entries are compared with time-varying threshold levels. In particular, instead of observing a subset of high-resolution entries of a low-rank matrix, we have access to a small number of one-bit samples, generated as a result of these comparisons. In order to recover the low-rank matrix using its coarsely quantized known entries, we begin by transforming the problem of one-bit matrix completion (one-bit MC) with time-varying thresholds into a nuclear norm minimization problem. The one-bit sampled information is represented as linear inequality feasibility constraints. We then develop the popular singular value thresholding (SVT) algorithm to accommodate these inequality constraints, resulting in the creation of the One-Bit SVT (OB-SVT). Our findings demonstrate that incorporating multiple time-varying sampling threshold sequences in one-bit MC can significantly improve the performance of the matrix completion algorithm. In pursuit of achieving this objective, we utilize diverse thresholding schemes, namely uniform, Gaussian, and discrete thresholds. To accelerate the convergence of our proposed algorithm, we introduce three variants of the OB-SVT algorithm. Among these variants is the randomized sketched OB-SVT, which departs from using the entire information at each iteration, opting instead to utilize sketched data. This approach effectively reduces the dimension of the operational space and accelerates the convergence. We perform numerical evaluations comparing our proposed algorithm with the maximum likelihood estimation method previously employed for one-bit MC, and demonstrate that our approach can achieve a better recovery performance.

翻译：探讨粗量化对矩阵补全的影响，研究极限场景下基于抖动的单比特感知，即矩阵元素与随时间变化的阈值电平进行比较。具体而言，我们不再直接观测低秩矩阵的部分高分辨率元素，而是通过比较过程获得少量单比特采样值。为了利用这些粗量化已知元素恢复低秩矩阵，首先将具有时变阈值的单比特矩阵补全问题转化为核范数最小化问题，其中单比特采样信息被表示为线性不等式可行性约束。随后，我们发展了流行奇异值阈值算法以兼容这些不等式约束，从而创建了单比特奇异值阈值算法（OB-SVT）。研究结果表明，在单比特矩阵补全中引入多种时变采样阈值序列可显著提升矩阵补全算法的性能。为实现这一目标，我们采用了均匀阈值、高斯阈值和离散阈值等不同阈值方案。为加速算法收敛，我们提出OB-SVT算法的三种变体，其中随机草图化OB-SVT变体摒弃了每次迭代使用全部信息的传统方式，转而采用草图化数据，有效降低了运算空间维度并加速收敛。通过数值评估，我们将所提算法与先前用于单比特矩阵补全的最大似然估计方法进行对比，证明我们的方法能实现更优的恢复性能。