Published Paper

Inserted: 23 feb 2018
Last Updated: 14 sep 2021

Journal: European Journal of Applied Mathematics
Volume: 31
Pages: 185-231
Year: 2018
Links: Published Paper

Abstract:

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 $\epsilon=\epsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $\epsilon_n\to 0$. The mathematical tool used to address this question is $\Gamma$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.

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• Cristoferi-Thorpe_-_Large_data_limit_for_a_phase_transition_model_with_the_p-Laplacian_on_point_clouds.pdf