Graph MBO as a semi-discrete implicit Euler scheme for graph Allen-Cahn flow

Abstract

In recent years there has been an emerging interest in PDE-like flows defined on finite graphs, with applications in clustering and image segmentation. In particular for image segmentation and semi-supervised learning Bertozzi and Flenner (2012) developed an algorithm based on the Allen-Cahn gradient flow of a graph Ginzburg-Landau functional, and Merkurjev, Kostić and Bertozzi (2013) devised a variant algorithm based instead on graph Merriman-Bence-Osher (MBO) dynamics.This work offers rigorous justification for this use of MBO in place of Allen-Cahn. First, we choose the double-obstacle potential for the Ginzburg-Landau functional, and derive existence, uniqueness and regularity results for the resulting graph Allen-Cahn flow. Next, we exhibit a "semi-discrete" time-discretisation scheme for Allen-Cahn of which MBO is a special case. We investigate the long-time behaviour of this scheme, and prove its convergence to the Allen-Cahn trajectory as the timestep vanishes. Finally, following a question raised in Van Gennip, Guillen, Osting and Bertozzi (2014), we exhibit results towards proving a link between doubleobstacle Allen-Cahn and mean curvature flow on graphs. We show some promising Γ-convergence results, and translate to the graph setting two comparison principles used in Chen and Elliott (1994) to prove the analogous link in the continuum.

Mass-conserving diffusion-based dynamics on graphs

“…First, we formulate on a graph the massconserving Allen-Cahn flow devised by Rubinstein and Sternberg [23], noticing that mass conservation continues to hold in the discrete setting. Next, following our earlier work in [11] and drawing on work in Van Gennip [14], we show that a formulation of a mass-conserving MBO scheme arises naturally as a special case of a semi-discrete scheme for the mass-conserving Allen-Cahn flow with the double-obstacle potential. We then examine various theoretical properties of this mass-conserving semi-discrete scheme.…”

“…• Extended the analysis in [11] to this new flow, proving a weak form, an explicit form, and uniqueness and existence theory for this flow (Theorems 3.6, 3.7, 3.8, and 3.9, respectively) and via the semi-discrete scheme proved that solutions exhibit monotonic decrease of the Ginzburg-Landau energy, and Lipschitz regularity (Theorems 5.8 and 5.10, respectively).…”

“…• Defined a mass-conserving semi-discrete scheme for this flow (Definition 4.1) and as in [11] proved that this scheme is equivalent to a variational scheme of which the MBO scheme is a special case (Theorems 4. 3 and 4.22).…”

“…• Following [11], derived a Lyapunov functional for the mass-conserving semi-discrete scheme (Theorem 4.23) and proved convergence of the scheme to the Allen-Cahn trajectory (Theorem 5.6), giving a novel proof of a key lemma from the method in [11].…”

An Empirical Study of Graph-Based Approaches for Semi-Supervised Time Series Classification

“…Details of the definition of the potential and the fidelity term incorporating the training data are found in [29]. Further extensions of this approach have been suggested in [41,13,7,15,16,18,10].…”