An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors

Yangyang Xu, Wotao Yin, Zaiwen Wen, and Yin Zhang

Published in Frontiers of Mathematics in China


This paper introduces a novel algorithm for the nonnegative matrix factorization and completion problem, which aims to find nonnegative matrices X and Y from a subset of entries Omega of a nonnegative matrix M so that XY approximates M.

 mathop{mathrm{minimize}}_{X,Y,Z}~|XY-Z|_F^2,~mathrm{subject to}~Xge 0,~yge 0,~Z_{i,j}=M_{i,j}~mathrm{for}~(i,j)inOmega.

This problem is closely related to the two existing problems: nonnegative matrix factorization and low-rank matrix completion, in the sense that it kills two birds with one stone. As it takes advantages of both nonnegativity and low rank, its results can be superior than those of the two problems alone. Our algorithm is applied to minimizing a non-convex constrained least-squares formulation and is based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties and numerical simulation results are presented.

Compared to a recent algorithm for nonnegative random matrix factorization, the proposed algorithm yields comparable factorization through accessing only half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the results of the proposed algorithm have overall better qualities than those of two recent algorithms for matrix completion.


Y. Xu, W. Yin, Z. Wen, and W. Yin, An alternating direction algorithm for matrix completion with nonnegative factors, Frontiers of Mathematics in China, 7(2), 365-384, 2012. DOI: 10.1007/s11464-012-0194-5

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