In this thesis we study the problem of exact completion for π Γ π sized matrix of rank π and the problem of low-rank estimation with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces. 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 π0-coherent column space matrices by

probability at least 1 β π using much smaller observations complexity than - πͺ(π0ππlogπ π )βthe 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 thesis, we provide experimental results that illustrate the strength of the algorithms proposed here. This thesis have been written mainly based on the paper [12].