Annals of Mathematical Sciences and Applications

Volume 1 (2016)

Number 1

Coordinate-friendly structures, algorithms and applications

Pages: 57 – 119

DOI: http://dx.doi.org/10.4310/AMSA.2016.v1.n1.a2

Authors

Zhimin Peng (Department of Mathematics, University of California at Los Angeles)

Tianyu Wu (Department of Mathematics, University of California at Los Angeles)

Yangyang Xu (Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minn., U.S.A.)

Ming Yan (Department of Computational Mathematics, Science and Engineering, and Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Wotao Yin (Department of Mathematics, University of California at Los Angeles)

Abstract

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.

Keywords

coordinate update, fixed point, operator splitting, primal-dual splitting, parallel, asynchronous

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