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 functional @math in the case @math has been considered by the second author and Theil in @cite_53 , where a similar @math -convergence result has been proved. The difference is that, in the case @math , the limit energy density function @math can be given explicitly, via an integral. In @cite_34 van Gennip and Bertozzi studied the Ginzburg-Landau functional on 4-regular graphs for @math and @math proving limits for @math and @math (both simultaneously and independently).
@cite_53 @cite_34
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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 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. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen--Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness @math and the mesh size of spatial discretization @math . Depending on their relative sizes, we obtain results in the framework of @math -convergence for the (i) subcritical ( @math ), (ii) critical ( @math ), and (iii) supercritical ( @math ) cases. The first case leads to the same area functional as the spatially continuous case while the third gives the same result as that coming from a ferromagnetic spin energy. The critical case can be regarded as an interpolation between the two.