We prove that the dynamics of the MBO scheme for data clustering converge to a viscosity solution to mean curvature flow. The main ingredients are (i) a new abstract convergence result based on quantitative estimates for heat operators and (ii) the derivation of these estimates in the setting of random geometric graphs. To implement the scheme in practice, two important parameters are the number of eigenvalues for computing the heat operator and the step size of the scheme. The results of the current paper give a theoretical justification for the choice of these parameters in relation to sample size and interaction width.
翻译:我们证明,MBO数据分组办法的动态与粘度解决方案相汇合,以表示曲线流。主要成份是:(一) 根据热操作员的定量估计得出的新的抽象趋同结果,以及(二) 在随机几何图的设定中得出这些估计数。为了在实践中实施这一办法,两个重要参数是计算热操作员的电子元值数和该办法的步数。本文的结果为选择这些参数与抽样大小和互动宽度相比提供了理论依据。