In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into submatrices which are filled with a proposed trimming step and then are recombined to form a low-rank completed matrix. The divide-and-conquer approach can significantly reduce computation complexity and storage requirement. Moreover, the proposed decomposition method can be naturally incorporated into any existing matrix completion methods to attain further gain. Unlike most existing approaches, the proposed method is not based on norm minimization nor SVD decomposition. This makes it possible to be applied beyond real domain and can be used in arbitrary fields including finite fields.

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Advances in DNA information extraction techniques have led to huge sequenced genomes from organisms spanning the tree of life. This increasing amount of genomic information requires tools for comparison of the nucleotide sequences. In this paper, we propose a novel nucleotide sequence alignment method based on sparse coding and belief propagation to compare the similarity of the nucleotide sequences. We used the neighbors of each nucleotide as features, and then we employed sparse coding to find a set of candidate nucleotides. To select optimum matches, belief propagation was subsequently applied to these candidate nucleotides. Experimental results show that the proposed approach is able to robustly align nucleotide sequences and is competitive to SOAPaligner [1] and BWA [2].

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