Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8
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The graph Merriman–Bence–Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in  gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme. In particular, these conditions only depend on local information at the node (information about neighbors and neighbors of neighbors of the node in question). In this paper we show that there does not exist any graph which satisfies these conditions. To make up for this negative result, this paper also presents positive results regarding the Merriman–Bence–Osher dynamics on star graphs and regular trees. In particular, we present sufficient (and in some cases necessary) results for the set membership of a given node to change in one iteration.
Keywordsgraph dynamics Merriman–Bence–Osher scheme threshold dynamics star graph regular tree graph
Mathematics Subject Classification (2010)Primary 35R02 49K15 Secondary 53C44 35K05 05C81
- 1.J. Budd and Y. van Gennip, Deriving graph MBO as a semi-discrete implicit Euler scheme for graph Allen–Cahn, towards a potential link to graph mean curvature flow, in preparation.Google Scholar
- 2.A. Chambolle, Total variation minimization and a class of binary MRF models, in: Energy Minimization Methods in Computer Vision and Pattern Recognition, LNCS Vol. 3757, pp. 136–152, Springer, 2005.Google Scholar
- 3.M. Cucuringu, A. Pizzoferrato, and Y. van Gennip, it An MBO scheme for clustering and semi-supervised clustering of signed networks, in preparation; preprint arXiv:1901.03091.
- 5.H. Hu, J. Sunu, and A.L. Bertozzi, Multi-class graph Mumford-Shah model for plume detection using the MBO scheme, in: Energy Minimization Methods in Computer Vision and Pattern Recognition, LNCS Vol. 8932, pp. 209–225, Springer, 2015.Google Scholar
- 7.B. Keetch and Y. van Gennip, A Max-Cut approximation using a graph based MBO scheme, to appear; preprint arXiv:1711.02419.
- 9.E. Merkurjev, J. Sunu, and A.L. Bertozzi, Graph MBO method for multiclass segmentation of hyperspectral stand-off detection video, in: 2014 IEEE International Conference on Image Processing (ICIP), pp. 689–693, IEEE, 2014.Google Scholar
- 10.E. Merkurjev, A.L. Bertozzi, and F.R.K. Chung, A semi-supervised heat kernel pagerank MBO algorithm for data classification, Tech. report, University of California, Los Angeles, 2016.Google Scholar
- 11.B. Merriman, J.K. Bence, and S. Osher, Diffusion generated motion by mean curvature, UCLA Department of Mathematics CAM report CAM 06–32, 1992.Google Scholar
- 12.B. Merriman, J.K. Bence, and S. Osher, Diffusion generated motion by mean curvature, in: AMS Selected Letters, Crystal Grower’s Workshop, pp. 73–83, AMS, 1993.Google Scholar
- 13.J. Szarski, Differential inequalities, PWN Warsaw, 1965.Google Scholar
- 15.Y. van Gennip, An MBO scheme for Minimizing the Graph Ohta–Kawasaki Functional, Journal of Nonlinear Science (2018); https://doi.org/10.1007/s00332-018-9468-8.
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