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 @math -Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of @math where @math is the number of data points. We study the large data asymptotics, i.e. as @math , in the regime where @math . The mathematical tool used to address this question is @math -convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.
The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds @math where @math denotes the number data points. In particular we show the limiting problem, in the sense of @math -convergence, is related to the total variation norm restricted to functions taking binary values; which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence. Author(s): Gennip, YV; Bertozzi, AL | Abstract: We study Γ-convergence of graph-based Ginzburg–Landau functionals, both the limit for zero diffusive interface parameter e → 0 and the limit for infinite nodes in the graph m → ∞. For general graphs we prove that in the limit e → 0 the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg–Landau functional. For both functionals we also study the simultaneous limit e → 0 and m → ∞, by expressing e as a power of m and taking m → ∞. Finally we investigate the continuum limit for a nonlocal means-type functional on a completely connected graph