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A Splitting Method for Orthogonality Constrained Problems

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Abstract

Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the constrained problems can be iteratively solved by computing the corresponding unconstrained problems and orthogonality constrained quadratic problems with analytic solutions. As applications, we demonstrate the robustness of our method in several problems including direction fields correction, noisy color image restoration and global conformal mapping for genus-0 surfaces construction. Numerical comparisons with existing methods are also conducted to illustrate the efficiency of our algorithms.

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Notes

  1. \(\varvec{F}^*\) is the standard pull-back map in differential geometry.

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Acknowledgments

Rongjie Lai’s work is supported by Zumberge Individual Award from USC’s James H. Zumberge Faculty Research and Innovation Fund. Stanley Osher’s work is supported by NSF grant DMS-1118971, DMS-0914561, ONR grant N00014-08-1-1119, an ARO MURI subcontract from University of South Carolina and an ARO MURI subcontract from Rice University. The authors would like to express their gratitude to Prof. Zaiwen Wen and Prof. Wotao Yin for sharing their curvilinear search codes for numerical comparisons.

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  1. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    Rongjie Lai

  2. Department of Mathematics, University of California, Los Angeles, CA, USA

    Stanley Osher

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Correspondence to Rongjie Lai.

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Lai, R., Osher, S. A Splitting Method for Orthogonality Constrained Problems. J Sci Comput 58, 431–449 (2014). https://doi.org/10.1007/s10915-013-9740-x

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