Matrix Completion from One-Bit Dither Samples

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 random dither 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 random dithering 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
O
ne-
B
it
SVT
(OBSVT). Our findings demonstrate that incorporating multiple random dither sequences in one-bit MC can significantly improve the performance of the matrix completion algorithm. In pursuit of achieving this objective, we utilize diverse dithering schemes, namely uniform, Gaussian, and discrete dithers. 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.