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Mass-conserving diffusion-based dynamics on graphs

An emerging technique in image segmentation, semi-supervised learning, and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012), which used the Allen-Cahn flow on a graph, and was then extended in Merkurjev, Kostic and Bertozzi (2013) using instead the Merriman-Bence-Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2019), we gave a theoretical justification for this use of the MBO scheme in place of Allen-Cahn flow, showing that the MBO scheme is a special case of a "semi-discrete" numerical scheme for Allen-Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992), we define a mass-conserving Allen-Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen-Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme. Finally, we exhibit initial work towards extending to the multi-class case, which in future work we seek to connect to recent work on multi-class MBO in Jacobs, Merkurjev and Esedoglu (2018).