3 research outputs found
Matrix Completion with Noise via Leveraged Sampling
Many matrix completion methods assume that the data follows the uniform
distribution. To address the limitation of this assumption, Chen et al.
\cite{Chen20152999} propose to recover the matrix where the data follows the
specific biased distribution. Unfortunately, in most real-world applications,
the recovery of a data matrix appears to be incomplete, and perhaps even
corrupted information. This paper considers the recovery of a low-rank matrix,
where some observed entries are sampled in a \emph{biased distribution}
suitably dependent on \emph{leverage scores} of a matrix, and some observed
entries are uniformly corrupted. Our theoretical findings show that we can
provably recover an unknown matrix of rank from just about
entries even when the few observed entries are corrupted with a
small amount of noisy information. Empirical studies verify our theoretical
results
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 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