We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

In this work we begin to rigorously analyze the MBO scheme for data clustering in the large data limit. Each iteration of the MBO scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of some dataset. For a subset of the nodes of the graph, the thresholding energy at time h is the amount of heat transferred from the subset to its complement at time h, rescaled by a factor √ h. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to minimizers of a suitably weighted optimal partition problem.

We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoglu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

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