  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 \textit{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 recovers $\mu_0$-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. At the end of the paper, we provide experimental results that
illustrate the strength of the algorithms proposed here.
