# MathSciDoc: An Archive for Mathematician ∫

#### Numerical Linear Algebramathscidoc:1912.43141

Frontiers of Mathematics in China, 7, (2), 365-384, 2012.4
This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices <i>X</i> and <i>Y</i> so that the product <i>XY</i> approximates a nonnegative data matrix <i>M</i> whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of <i>M</i> are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classical alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to
```@inproceedings{yangyang2012an,
title={An alternating direction algorithm for matrix completion with nonnegative factors},
author={Yangyang Xu, Wotao Yin, Zaiwen Wen, and Yin Zhang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112518167965701},
booktitle={Frontiers of Mathematics in China},
volume={7},
number={2},
pages={365-384},
year={2012},
}
```
Yangyang Xu, Wotao Yin, Zaiwen Wen, and Yin Zhang. An alternating direction algorithm for matrix completion with nonnegative factors. 2012. Vol. 7. In Frontiers of Mathematics in China. pp.365-384. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112518167965701.