# American Institute of Mathematical Sciences

November  2016, 10(4): 1149-1180. doi: 10.3934/ipi.2016036

## A minimal surface criterion for graph partitioning

 1 Montana State University, Department of Mathematical Sciences, 2-214 Wilson Hall, Box 172400, Bozeman, MT 59717-2400, United States 2 University of Utah, Salt Lake City, Department of Mathematics, 155 S. 1400 E, Rm 233, Salt Lake City, UT 84112, United States

Received  July 2015 Revised  May 2016 Published  October 2016

We consider a geometric approach to graph partitioning based on the graph Beltrami energy, a discrete version of a functional that appears in classical minimal surface problems. More specifically, the optimality criterion is given by the sum of the minimal Beltrami energies of the partition components. Since the Beltrami energy interpolates between the Total Variation and Dirichlet energies, various results for optimal partitions for these two energies can be recovered. We adapt primal-dual convex optimization methods to solve for the minimal Beltrami energy for each component of a given partition. A rearrangement algorithm is proposed to find the graph partition to minimize a relaxed version of the objective. The method is applied to several clustering problems on graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and image segmentation. The model has a semisupervised extension and provides a natural representative for the clusters as well.
Citation: Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036
##### References:
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##### References:
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Merkurjev, A. L. Bertozzi, A. Flenner and A. G. Percus, Multiclass data segmentation using diffuse interface methods on graphs, IEEE Transactions on Pattern Analysis and Machine Intelligence, 36 (2014), 1600-1613. doi: 10.1109/TPAMI.2014.2300478.  Google Scholar [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Springer Science & Business Media, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar [31] T. C. Hales, The honeycomb conjecture, Discrete & Computational Geometry, 25 (2001), 1-22. doi: 10.1007/s004540010071.  Google Scholar [32] B. Helffer, On spectral minimal partitions: A survey, Milan J. Math., 78 (2010), 575-590. doi: 10.1007/s00032-010-0129-0.  Google Scholar [33] B. Helffer and T. Hoffmann-Ostenhof, Remarks on two notions of spectral minimal partitions, Adv. Math. Sci. Appl., 20 (2010), 249-263.  Google Scholar [34] B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, On spectral minimal partitions: The case of the sphere, 2010, Around the research of Vladimir Maz'ya. III, Int. Math. Ser. (N. Y.), Springer, New York, 13 (2010), 153-178. doi: 10.1007/978-1-4419-1345-6_6.  Google Scholar [35] C. Herring, Some theorems on the free energies of crystal surfaces, Physical Review, 82 (1951), 87-93. doi: 10.1103/PhysRev.82.87.  Google Scholar [36] R. Kaftory, N. A. Sochen and Y. Y. Zeevi, Variational blind deconvolution of multi-channel images, Int. J. Imaging Syst. Technol., 15 (2005), 56-63. doi: 10.1002/ima.20038.  Google Scholar [37] R. Kimmel, R. Malladi and N. Sochen, Images as embedded maps and minimal surfaces: Movies, color, texture, and volumetric medical images, Int. J. Comput. Vis., 39 (2000), 111-129. Google Scholar [38] C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing, 17 (2008), 1940-1949. doi: 10.1109/TIP.2008.2002304.  Google Scholar [39] Z. Liang and Y. Li, Beltrami flow in Hilbert space with applications to image denoising, Journal of Electronic Imaging, 21 (2012), 043019, 1-11. doi: 10.1117/1.JEI.21.4.043019.  Google Scholar [40] S. Lloyd, Least squares quantization in PCM, IEEE Transactions on Information Theory, 28 (1982), 129-137. doi: 10.1109/TIT.1982.1056489.  Google Scholar [41] L. Lopez-Perez, R. Deriche and N. Sochen, The Beltrami flow over triangulated manifolds, in CVAMIA and MMBIA Workshop at ECCV 2004, 3117 (2004), 135-144. doi: 10.1007/978-3-540-27816-0_12.  Google Scholar [42] E. Merkurjev, T. Kostic and A. L. Bertozzi, An MBO scheme on graphs for segmentation and image processing, SIAM J. Imaging Sciences, 6 (2013), 1903-1930. doi: 10.1137/120886935.  Google Scholar [43] B. Merriman, J. K. Bence and S. Osher, Diffusion Generated Motion by Mean Curvature, Technical report, UCLA CAM Report 92-18, 1992. Google Scholar [44] B. Merriman, J. K. Bence and S. Osher, Diffusion generated motion by mean curvature,, AMS Selected Letters, (): 73.   Google Scholar [45] B. Osting and C. D. White, Nonnegative matrix factorization of transition matrices via eigenvalue optimization, in NIPS OPT, 2013. Google Scholar [46] B. Osting, C. D. White and É. Oudet, Minimal Dirichlet energy partitions for graphs, SIAM Journal on Scientific Computing, 36 (2014), A1635-A1651. doi: 10.1137/130934568.  Google Scholar [47] É. Oudet, Approximation of partitions of least perimeter by $\Gamma$-convergence: around Kelvin's conjecture, Experimental Mathematics, 20 (2011), 260-270. doi: 10.1080/10586458.2011.565233.  Google Scholar [48] J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis Aux Seules Forces Moléculaires, Gauthier-Villars, Paris, 1873. Google Scholar [49] G. Pólya and G. 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