论文标题
图上基于质量的扩散动力学
Mass-conserving diffusion-based dynamics on graphs
论文作者
论文摘要
图像分割，半监督学习和一般分类问题中的一种新兴技术涉及在有限图上定义的相分离流的使用。该技术是在Bertozzi和Flenner（2012）中启用的，该技术在图上使用了Allen-Cahn流，然后在Merkurjev，Kostic和Bertozzi（2013）中使用Merriman-Bence-berce-osher（MBO）方案延伸。在作者Budd and van Gennip（2019）的先前工作中，我们为这种使用MBO方案代替Allen-Cahn流提供了理论理由，这表明MBO方案是Allen-Cahn流的“半污物”数值方案的特殊情况。 在本文中，我们扩展了这项较早的工作，表明通过半混凝土方案的这种链接可以强大地传递给质量支持案例。受Rubinstein和Sternberg（1992）的启发，我们在图上定义了质量支持的艾伦·卡纳方程。然后，借助凸优化的工具，我们表明我们的早期机器可以在图表上推导质量支持的MBO方案，以作为用于质量支持的艾伦·卡恩（Allen-Cahn）的半差异方案的特殊情况。我们对这种流程和方案进行了理论分析，证明了该方案的流动和收敛性的各种所需属性，也证明了半分化方案为质量支持MBO方案的解决方案提供了选择功能。最后，我们展示了扩展到多级案例的初步工作，在未来的工作中，我们试图与Jacobs，Merkurjev和Esedoglu（2018）中的多级MBO进行连接。
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).
