Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 18-CNA-004 Large Data Limit for a Phase Transition Model with the $p$-Laplacian on Point Clouds R. CristoferiDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh PA 15213-3890 USArcristof@andrew.cmu.edu Matthew ThorpeDepartment of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge, UKAbstract: 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 $\varepsilon$ = $\varepsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n \to \infty$, in the regime where $\varepsilon_n \to 0$. 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.Get the paper in its entirety as  18-CNA-004.pdf