Optimal Adaptive Matrix Completion

We study the problem of exact completion for $m \times n$ sized matrix of rank r with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call sparsity-number here). Using this relation, we propose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recover $\mu_0$-coherent column space matrices by probability at least $1-\epsilon$ using much smaller observations complexity than - $\mathcal{O}(\mu_0 rn \mathrm{log}\frac{r}{\epsilon})$ - the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, which requires minimal pre-information as either row or column space is not being highly coherent. We provide an extension of these algorithms that is robust to sparse random noise. Besides, we propose an additional low-rank estimation algorithm that is robust to any small noise by adaptively studying the shape of column space. At the end of the paper, we provide experimental results that illustrate the strength of the algorithms proposed here.