点云上p-Laplacian相变模型的大数据限
Large data limit for a phase transition model with the p-Laplacian on point clouds
摘要：
研究了用于数据分类和聚类的非局部各向异性Ginzburg-Landau型泛函的相容性。Ginzburg-Landau目标泛函结合了双势阱和$p$ -Laplacian算子，前者有利于指示值函数，后者则加强了规则性。在两项之间的适当缩放下，极小值显示出$ε=ε_n$量级的相变，其中$n$是数据点的数量。我们研究了大数据的渐近性，即作为$n\to \infty$，在该制度中，$ε_n\to 0$。用于解决这个问题的数学工具是$Γ$ -收敛。特别地，证明了该离散模型收敛于加权各向异性周长。
英文摘要：
The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $ε=ε_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $ε_n\to 0$. The mathematical tool used to address this question is $Γ$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.
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