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：我们探讨了粗量化对矩阵补全的影响，特别是在抖动单比特感知的极端场景下，矩阵条目与随时间变化的阈值水平进行比较。具体而言，我们不再观测低秩矩阵高分辨率条目的子集，而是通过这些比较结果获得少量的单比特样本。为了利用这些粗量化的已知条目恢复低秩矩阵，我们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。单比特采样信息被表示为线性不等式可行性约束。随后，我们扩展了流行的奇异值阈值算法以容纳这些不等式约束，从而创建了单比特奇异值阈值算法。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以显著提升矩阵补全算法的性能。为实现这一目标，我们采用了多种阈值方案：均匀阈值、高斯阈值和离散阈值。为加速所提算法的收敛，我们引入了单比特奇异值阈值算法的三种变体。其中一种变体是随机草图化单比特奇异值阈值算法，该算法在每次迭代中不再使用全部信息，而是利用草图化数据。这种方法有效降低了操作空间的维度并加速了收敛。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了比较，结果表明我们的方法能够实现更好的恢复性能。