publication . Other literature type . Preprint . Article . 2020

A Vectorial Approach to Unbalanced Optimal Mass Transport

Jiening Zhu; Rena Elkin; Jung Hun Oh; Joseph O. Deasy; Allen Tannenbaum;
Open Access English
  • Published: 06 Aug 2020
Unbalanced optimal mass transport (OMT) seeks to remove the conservation of mass constraint by adding a source term to the standard continuity equation in the Benamou-Brenier formulation of OMT. In this study, we show how the unbalanced case fits into the vector-valued OMT framework simply by adding an auxiliary source layer and taking the flow between the source layer and the original layer(s) as the source term. This allows for unbalanced models both in the scalar and vector-valued density settings. The results are demonstrated on a number of synthetic and real vector-valued data sets.
free text keywords: Article, Mathematics - Optimization and Control, Mathematical Physics, Mathematics - Functional Analysis, General Engineering, General Materials Science, General Computer Science
Funded by
NIH| Glymphatic function in a transgenic rat model of Alzheimer's disease
  • Funder: National Institutes of Health (NIH)
  • Project Code: 5R01AG048769-04
  • Funder: National Institutes of Health (NIH)
  • Project Code: 2P30CA008748-43
Download fromView all 3 versions
Europe PubMed Central
Other literature type . 2020
IEEE Access
Article . 2020
Provider: Crossref
16 references, page 1 of 2

[1] Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein GAN. 2017.

[2] Jean-David Benamou and Yann Brenier. A computational uid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3):375{ 393, 2000.

[3] Yongxin Chen, Tryphon Georgiou, and Allen Tannenbaum. Wasserstein Geometry of Quantum States and Optimal Transport of Matrix-Valued Measures, pages 139{150. 01 2018.

[4] Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Interpolation of density matrices and matrix-valued measures: The unbalanced case. Euro. Jnl of Applied Mathematics, 30(3):458{480, 2018.

[5] Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Matrix optimal mass transport: a quantum mechanical approach. IEEE Trans. Automatic Control, 63(8):2612 { 2619, 2018.

[6] Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Vector-valued optimal mass transport. SIAM Journal Applied Mathematics, 78(3):1682{1696, 2018.

[7] Yongxin Chen, Kaoru Yamamoto, Eldad Haber, Tryphon T. Georgiou, and Allen Tannenbaum. An e cient algorithm for matrix-valued and vector-valued optimal mass transport. in preparation, 2017.

[8] Lenaic Chizat, Gabriel Peyre, Bernhard Schmitzer, and Francois-Xavier Vialard. Unbalanced Optimal Transport: Geometry and Kantorovich Formulation. working paper or preprint, August 2015.

[9] Lenaic Chizat, Gabriel Peyre, Bernhard Schmitzer, and Francois-Xavier Vialard. An interpolating distance between optimal transport and Fisher-Rao metrics. Foundations of Computational Mathematics, 10:1{44, 2016.

[10] Wilfrid Gangbo, Wuchen Li, Stanley Osher, and Michael Puthawala. Unnormalized optimal transport. Journal of Computational Physics, 399:1{19, 2019.

[11] Steven Haker, Lei Zhu, Allen Tannenbaum, and Sigurd Angenent. Optimal mass transport for registration and warping. International Journal of Computer Vision, 60(3):225{240, 2004.

[12] James C. Mathews, Saad Nadeem, Maryam Pouryahya, Zehor Belkhatir, Joseph O. Deasy, Arnold J. Levine, and Allen R. Tannenbaum. Functional network analysis reveals an immune tolerance mechanism in cancer. Proceedings of the National Academy of Sciences, 117(28):16339{16345, jul 2020. [OpenAIRE]

[13] Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Di erential Equations, 2001.

[14] Svetlozar T Rachev and Ludger Ruschendorf. Mass Transportation Problems: Volumes I and II. Springer Science & Business Media, 1998.

[15] Cedric Villani. Topics in Optimal Transportation. American Mathematical Soc., 2003.

16 references, page 1 of 2
Any information missing or wrong?Report an Issue