Projected Gradient Method Combined with Homotopy Techniques for Volume-Measure-Preserving Optimal Mass Transportation Problems

Mei-Heng Yueh National Taiwan Normal University Tsung-Ming Huang Tiexiang Li Wen-Wei Lin Shing-Tung Yau

Computational Geometry mathscidoc:2006.09001

Optimal mass transportation has been widely applied in various fields, such as data compression, generative adversarial networks, and image processing. In this paper, we adopt the projected gradient method, combined with the homotopy technique, to nd a minimal volume-measure-preserving solution for a 3-manifold optimal mass transportation problem. The proposed projected gradient method is shown to be sublinearly convergent at a rate of O(1/k). Several numerical experiments indicate that our algorithms are able to signi cantly reduce transportation costs. Some applications of the optimal mass transportation maps - to deformations and canonical normalizations between brains and solid balls - are demonstrated to show the robustness of our proposed algorithms.
Optimal Mass Transportation
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@inproceedings{mei-hengprojected,
  title={Projected Gradient Method Combined with Homotopy Techniques for Volume-Measure-Preserving Optimal Mass Transportation Problems},
  author={Mei-Heng Yueh, Tsung-Ming Huang, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200624141205342115700},
}
Mei-Heng Yueh, Tsung-Ming Huang, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau. Projected Gradient Method Combined with Homotopy Techniques for Volume-Measure-Preserving Optimal Mass Transportation Problems. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200624141205342115700.
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