Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs.

Area of Science:

• Computational mathematics
• Computer vision
• Data science

Background:

• Multiclass segmentation of high-dimensional data on graphs is a challenging problem.
• Existing methods often struggle with complex data structures and high dimensionality.

Purpose of the Study:

• To introduce two novel graph-based algorithms for multiclass segmentation of high-dimensional data.
• To adapt diffuse interface models and Gibbs simplex for effective multiclass segmentation.

Main Methods:

• Developed two algorithms: one minimizing a modified Ginzburg-Landau functional via convex splitting, and another using a graph-based Merriman-Bence-Osher (MBO) scheme.
• Utilized efficient numerical solvers for graph Laplacian eigenvalues/eigenvectors and leveraged matrix sparsity.
• Extended Ginzburg-Landau functional with a Gibbs simplex for multiclass partitioning.

Main Results:

• Demonstrated algorithm performance on synthetic data, image labeling, and benchmark datasets (MNIST, COIL, WebKB).
• Achieved results competitive with or superior to current state-of-the-art multiclass graph segmentation methods.
• Validated the effectiveness of the convex splitting and MBO-based approaches for high-dimensional graph data.

Conclusions:

• The proposed graph-based algorithms provide effective solutions for multiclass segmentation of high-dimensional data.
• The adapted diffuse interface model and Gibbs simplex approach show significant promise.
• These methods represent a valuable advancement in graph-based data analysis and segmentation.