In the era of big data, the matrix completion (MC) problem has become increasingly popular in machine learning and data mining. Many algorithms, such as singular value thresholding, soft-impute and fixed point continuation, have been proposed for solving this problem. Typically, these existing algorithms require implementing a singular value decomposition of a data matrix at each iteration. Thus, these algorithms are not scalable when the size of the matrix is very large. Motivated by the principle of robust principal component analysis, in this paper we propose a novel MC algorithm, called robust and scalable MC with Kronecker product (RSKP), which models the original data matrix as a low-rank matrix plus a sparse matrix. Furthermore, we represent the low-rank matrix as the Kronecker product of two small-size matrices. Using the Kronecker product makes the model scalable, and introducing the sparse matrix makes the model more robust. We apply our RSKP algorithm to image recovery problems which can be naturally represented by a data matrix with the Kronecker product structure. Experimental results show that our RSKP is efficient and effective in real applications.
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