Large data limit for a phase transition model with thep-Laplacian on point clouds

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 functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of ε = ε n where n is the number of data points. We study the large data asymptotics, i.e. as n → ∞, in the regime where ε n → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter. procedure in order to obtain some desired features of the classification. Furthermore, understanding the large data limits can open up new algorithms. This paper is part of an ongoing project aimed at justifying analytically the consistency of several models for soft labelling used by practitioners. Here we consider a generalization of the approach introduced by Bertozzi and Flenner in [7] (see also [17], for an introduction on this topic see [56]), where a Ginzburg-Landau (or Modica-Mortola, see [40,41]) type functional is used as the underlining energy to minimize in the context of the soft classification problem. The functional we consider is a discretisation of the non-local Ginzburg-Landau functional studied by Alberti and Bellettini [1,2] with the generalisation that we consider non-uniform densities and p (rather than 2 ) cost on finite differences. Our goal is to prove the consistency of the model.There are multiple extensions to the approach we consider here; for instance our Ginzburg-Landau functional is based on the p-Laplacian, one can also consider the normalised p-Laplacian or the random walk Laplacian (see [42,47]). Further open problems concern the extention to multi-phase labelling (see [30]) and convergence of the associated gradient flows.The paper is organized as follows: in the following subsection we define the discrete model, and in Subsection 1.2 we define the continuum limiting problem. The main results are given in Section 1.3 with the proofs presented in Sections 4 and 5. In Section 1.4 we give an overview on the related literature. In Sections 1.5 and 1.6 we include two examples with the purpose of demonstrating key properties of our functional; in particular how the choice of p effects minimizers of our Ginzburg-Landau functional and an example to image segmentation. Section 2 contains some prelimimary material we include for the convenience of the reader. Finally, Section 3 is devoted to the proofs of some technical results that are of interest in their own right, and are later used in the proofs in Section 4.The consistency of the model is studied using Γ-convergence (see Section 2.4), a very important tool introduced by De Giorgi in the 70's to understand the limiting behavior of a sequence of functionals (see [21]). This kind of variational convergence gives, almost immediately, convergence of minimizers.

Continuous-domain assignment flows

“…Since the assignment flow returns image partitions when applied to image features on a grid graph, the situation reminds us of the Mumford-Shah functional [23] -more precisely: its restriction to piecewise constant functions -and its approximation by a sequence ofconverging smooth elliptic problems [4]. Likewise, one may regard the concave second term of (4.18) together with the convex constraint S ∈ A 1 g as a vector-valued counterpart of the basic nonnegative double-well potential of scalar phase-field models for binary segmentation [28,12]. In these works, too, non-smooth limit cases result from -converging simpler problems.…”

Large data and zero noise limits of graph-based semi-supervised learning algorithms

“…Let Assumptions 1-3 hold. For α ≥ 1 and τ ≥ 0, consider the functional J p with Labelling Model 1 defined by (15). Then, the functional J p has a unique minimizer in H α (Ω).…”

“…The statement of the probit algorithm in the context of graph based semi-supervised learning may be found in [6]. An approach bridging the combinatorial and Gaussian process approaches is the use of Ginzburg-Landau models which work with real numbers but use a penalty to constrain to values close to the range of the label data {±1}; these methods were introduced in [4], large data limits studied in [15,42,44], and given a probabilistic interpretation in [6]. Finally we mention the Bayesian level set method.…”

A Compactness Theorem for functions on Poisson point clouds

“…The novelty of this work relies exactly in the assumption on the decay of s ε in terms of ε, expressed by (1.2) with 0 < β < +∞. The typical approach for energies defined on point clouds (as for instance in [20], [23], [34], [35]) makes use of the hypothesis β = +∞ in order to study the asymptotic behavior as ε → 0. In the long-range regime β = +∞, the topology of convergence is the strong one induced by the T L 1 distance (see for instance [25] or [35]).…”