Coordinate Friendly Structures, Algorithms and Applications

Zhimen Peng, Tianyu Wu, Yangyang Xu, Ming Yan, and Wotao Yin

Published in Annals of Mathematical Sciences and Applications

Overview

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 subproblems. 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 such as image deblurring, portfolio optimization, second-order cone programming, as well as matrix decomposition. 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.

Citation

Z. Peng, T. Wu, Y. Xu, M. Yan, and W. Yin, Coordinate Friendly Structures, Algorithms and Applications, Annals of Mathematical Sciences and Applications 1(1), 57-119, 2016. DOI: 10.4310/AMSA.2016.v1.n1.a2


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