Graph Merriman--Bence--Osher as a SemiDiscrete Implicit Euler Scheme for Graph Allen--Cahn Flow

Budd, Jeremy; Gennip, Yves van

doi:10.1137/19m1277394

Cited by 11 publications

(99 citation statements)

References 23 publications

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“…This work extends the work in [13] in four key ways. First, introducing fidelity forcing changes the character of the dynamics, for example, making graph diffusion affine, which changes a number of results/proofs, and it is thus of interest that the SDIE link continues to hold between the MBO scheme and the ACE.…”

Section: Introductionmentioning

confidence: 52%

“…In this paper, we investigate the Allen‐Cahn gradient flow of the Ginzburg‐Landau functional on a graph, and the Merriman‐Bence‐Osher (MBO) scheme on a graph, with fidelity forcing. We extend the definition of the semi‐discrete implicit Euler (SDIE) scheme (introduced in [13] for the graph Allen‐Cahn equation (ACE)) to the case of fidelity forcing, and prove that the key results of [13] hold true in the fidelity forced setting, that isthe MBO scheme with fidelity forcing is a special case of the SDIE scheme with fidelity forcing; and the SDIE solution converges to the solution of Allen‐Cahn with fidelity forcing as the SDIE time step tends to zero. …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this paper we have:Defined a double‐obstacle ACE with fidelity forcing (Definition 2.6), and extended the theory of [13] to this equation (Theorem 2.7). Defined an SDIE scheme for this ACE (Definition 2.8) and following [13] proved that this scheme is a generalization of the fidelity forced MBO scheme (Theorem 2.9), derived a Lyapunov functional for the SDIE scheme (Theorem 2.12), and proved that the scheme converges to the ACE solution as the time‐step tends to zero (Theorem 2.17). Described how to employ the SDIE scheme as a generalization of the MBO‐based classification algorithm in [32]. Developed a method, inspired by [3], using the QR decomposition to extract an approximate SVD of the normalized graph Laplacian from the Nyström extension (Algorithm 1), which avoids the potential for errors in the method from [6, 32] that can arise from taking the square root of a non‐positive‐semi‐definite matrix, and empirically produces much better performance than the [6, 32] method (Figure 4) in accuracy, stability, and speed. Developed a method using the quadratic error Strang formula for matrix exponentials [39] for computing fidelity forced graph diffusion (Algorithm 2), which empirically incurs a lower error than the error incurred by the semi‐implicit Euler method used in [32] (Figure 6), and explored other techniques with the potential to further reduce error (Table 1).…”

Section: Introductionmentioning

confidence: 99%

“…In [13] it was shown that the graph MBO scheme without fidelity forcing could be obtained as a special case of a SDIE scheme for the ACE (without fidelity forcing), if the smooth double‐well potential was replaced by the double‐obstacle potential defined in (), and that solutions to the SDIE scheme converge to the solution of the graph ACE as the time step converges to zero. This double‐obstacle potential was studied for the continuum ACE in [8‐10] and was used in the graph context in [11].…”

Section: Introductionmentioning

confidence: 99%

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Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

2021

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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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Section: Introductionmentioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

2021

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Multiscale laplacian learning

2022

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Mass-conserving diffusion-based dynamics on graphs

Budd¹,

Gennip²

2021

Eur. J. Appl. Math

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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 Simulation10(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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Graph Merriman--Bence--Osher as a SemiDiscrete Implicit Euler Scheme for Graph Allen--Cahn Flow

Cited by 11 publications

References 23 publications

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

Multiscale laplacian learning

Mass-conserving diffusion-based dynamics on graphs

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