Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs

Abstract

Abstract. In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.Mathematics Subject Classification (2010). 34B45, 35R02, 53C44, 53A10, 49K15, 49Q05, 35K05.

Nonlinear Operators on Graphs via Stacks

“…On the other hand, some regularisation techniques and nonlinear operators have been introduced on function evaluated on graph via directional derivative [10] [11] [12] or discrete version of the p-Laplacian [13]. We adopt a different viewpoint, our approach is inspired from the signal processing approach on graphs [14,15,16,17]. Thus, we firstly review graph signal decomposition by upper-level sets, convolution and diffusion on graphs, and then we present a general formulation of flat and non-flat morphology on graphs, a family of nonlinear transformations and its connection to mean curvature motion on graphs [18,17,19].…”

“…We adopt a different viewpoint, our approach is inspired from the signal processing approach on graphs [14,15,16,17]. Thus, we firstly review graph signal decomposition by upper-level sets, convolution and diffusion on graphs, and then we present a general formulation of flat and non-flat morphology on graphs, a family of nonlinear transformations and its connection to mean curvature motion on graphs [18,17,19]. Finally, the experimental section includes some examples to illustrate the interest of our method.…”

Convergence of the Graph Allen–Cahn Scheme

“…The diffuse interface models can often be used as a proxy for total variation (TV) minimization since the -limit of the Ginzburg-Landau functional is shown to be the TV semi-norm [20]. The key observation linking the two areas above is that the TV semi-norm, when suitably generalized to weighted graphs, coincides with the graph cut functional for discrete valued functions on graphs [29]. Hence techniques for TV minimization can also be applied to solve the graph cut problem.…”

Introduction: Big data and partial differential equations

“…This suggests that graph curvature and graph mean curvature flow are interesting concepts to consider as well. In [34], the authors introduced both. The graph curvature of a node set S is given by…”

“…In [34], the authors started studying the very interesting question whether the graph Allen-Cahn equation, graph MBO scheme and graph mean curvature flow are as intimately connected as their continuum counterparts, but establishing such connections is still mostly an open problem.…”