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

This paper introduces a semi-discrete implicit Euler (SDIE) scheme for the Allen-Cahn equation (ACE) with fidelity forcing on graphs. Bertozzi and Flenner (2012) pioneered the use of this differential equation as a method for graph classification problems, such as semi-supervised learning and image segmentation. In Merkurjev, Kosti\'c, and Bertozzi (2013), a Merriman-Bence-Osher (MBO) scheme with fidelity forcing was used instead, as the MBO scheme is heuristically similar to the ACE. This 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 using the double-obstacle potential in the ACE, as was shown in Budd and Van Gennip (2020) for ACE without fidelity forcing. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the SDIE time step tends to zero. Next, 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 Nystr\"om extension, which outperforms the method used in e.g. Bertozzi and Flenner (2012) in accuracy, stability, and speed. Moreover, we replace the Euler discretisation 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 of the SDIE and MBO schemes. We find that whilst 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 Nystr\"om extension.