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.
This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit @math for ( : [0, ) [0, ) ) continuous and (( ) L^1( R ^N) ) converging to ( ), is related to the weak regularity of (f L^1_ loc ( ) ). This approach gives an alternative way of defining the Sobolev spaces W 1,p . We also briefly discuss the ( )-convergence of (1) with respect to the (L^1( ) )-topology. The standard seminorm in the space @math , with @math < @math , does not converge, when @math approaches @math , to the corresponding @math seminorm. We prove that continuity is restored provided we multiply the @math seminorm by an appropriate factor.  where u is a scalar density function on a domain of R and takes values in [−1, 1], W is a positive double-well potential which vanishes at ±1, and J is a positive, possibly anisotropic, interaction potential which vanishes at infinity (see paragraph 1.2 for precise definitions). The scalar function u represents the macroscopic density profile of a system which has two equilibrium pure phases described by the profiles u ≡ +1 and u ≡ −1. The integral ∫ W (u) at the right side of (1.1) forces a minimizer of F to take values close to +1 and −1 (phase separation), while the double integral represents an interaction energy which penalizes the spatial inhomogeneity of the system (surface tension). In equilibrium Statistical Mechanics functionals of the form (1.1) arise as free energies of continuum limits of Ising spin systems on lattices; in this setting u plays the role of a macroscopic magnetization density and J is a ferromagnetic Kac potential (see for instance [2] and references therein). We underline the analogy with the more familiar gradient theory for phase transition proposed in [9], where the free energy of the system is of the form In this paper we consider a nonlocal anisotropic model for phase separation in two-phase fluids at equilibrium, and show that when the thickness of the interface tends to zero in a suitable way, the classical surface tension model is recovered. Relevant examples are given by continuum limits of ferromagnetic Ising systems in equilibrium statistical mechanics. 