A large number of modern learning problems involve working with highly interrelated and interconnected data. Graph-based learning is an emerging technique for approaching such problems, by representing this data as a graph (a.k.a. a network). That is, the points of data are represented by the vertices of the graph, and then the edges linking these vertices represent the relationships between the points of data. This provides a unified perspective for thinking about all sorts of interrelated data: the vertices could represent pixels in an image or people in a social network, and the underlying framework would be the same... @en

This paper introduces a semi-discrete implicit Euler (SDIE) scheme for the Allen-Cahn equation (ACE) with fidelity forcing on graphs. The continuous-in-time version of this differential equation was pioneered by Bertozzi and Flenner in 2012 as a method for graph classification problems, such as semi-supervised learning and image segmentation. In 2013, Merkurjev et. al. used a Merriman-Bence-Osher (MBO) scheme with fidelity forcing instead, as heuristically it was expected to give similar results to the ACE. The current paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires the use of the double-obstacle potential in the ACE, as was already demonstrated by Budd and Van Gennip in 2020 in the context of ACE without a fidelity forcing term. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the discrete time step converges to zero. In the second part of the paper we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used by, for example, Bertozzi and Flenner in 2012, in accuracy, stability, and speed. Moreover, we replace the Euler discretization for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance with that of the graph MBO scheme with fidelity forcing. We find that while the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

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An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation 10(3), 1090-1118), which used the Allen-Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci. 6(4), 1903-1930) using instead the Merriman-Bence-Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal. 52(5), 4101-4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen-Cahn flow, showing that the MBO scheme is a special case of a 'semi-discrete' numerical scheme for Allen-Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math. 48, 249-264), we define a mass-conserving Allen-Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen-Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

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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 semisupervised learning Bertozzi and Flenner [Multiscale Model. Simul., 10 (2012), pp. 1090--1118] developed an algorithm based on the Allen--Cahn (AC) gradient flow of a graph Ginzburg--Landau functional, and Merkurjev, Kostić, and Bertozzi [SIAM J. Imaging Sci., 6 (2013), pp. 1903--1930] devised a variant algorithm based instead on graph Merriman--Bence--Osher (MBO) dynamics. This work offers rigorous justification for this use of the MBO scheme in place of AC flow. First, we choose the double-obstacle potential for the Ginzburg--Landau functional and derive well-posedness and regularity results for the resulting graph AC flow. Next, we exhibit a “semidiscrete” time-discretization scheme for AC flow of which the MBO scheme is a special case. We investigate the long-time behavior of this scheme and prove its convergence to the AC trajectory as the time-step vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting, and Bertozzi [Milan J. Math., 82 (2014), pp. 3--65], we exhibit results toward proving a link between double-obstacle AC flow and mean curvature flow on graphs. We show some promising $\Gamma$-convergence results and translate to the graph setting two comparison principles used by Chen and Elliott [Proc. Math. Phys. Sci., 444 (1994), pp. 429--445] to prove the analogous link in the continuum.


Read More: https://epubs.siam.org/doi/10.1137/19M1277394@en