Coordinate-friendly structures, algorithms and applications

Zhimin Peng UCLA Tianyu Wu UCLA Yangyang Xu University of Alabama Ming Yan Michigan State University Wotao Yin UCLA

Numerical Analysis and Scientific Computing Optimization and Control Machine Learning mathscidoc:1703.25025

Annals of Mathematical Sciences and Applications, 1, (1), 57 – 119, 2016
This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.
coordinate update, fixed point, operator splitting, primal-dual splitting, parallel, asynchronous
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@inproceedings{zhimin2016coordinate-friendly,
  title={Coordinate-friendly structures, algorithms and applications},
  author={Zhimin Peng, Tianyu Wu, Yangyang Xu, Ming Yan, and Wotao Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314100319961226694},
  booktitle={Annals of Mathematical Sciences and Applications},
  volume={1},
  number={1},
  pages={57 – 119},
  year={2016},
}
Zhimin Peng, Tianyu Wu, Yangyang Xu, Ming Yan, and Wotao Yin. Coordinate-friendly structures, algorithms and applications. 2016. Vol. 1. In Annals of Mathematical Sciences and Applications. pp.57 – 119. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314100319961226694.
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