Heat Kernel

The heat kernel, a function describing heat diffusion on a manifold or graph, is a fundamental tool in various fields, primarily used for analyzing data's underlying geometric structure and performing tasks like shape analysis and graph classification. Current research focuses on extending its applications through novel algorithms, such as continuous product graph neural networks and Finsler-Laplace-Beltrami operators, which leverage the heat kernel's properties for improved efficiency and accuracy in handling complex data structures like multiple interacting graphs and anisotropic manifolds. These advancements are significantly impacting machine learning on graphs, shape analysis, and other areas by providing more robust and efficient methods for data embedding, feature extraction, and classification. The heat kernel's inherent connection to diffusion processes also allows for the development of novel spectral algorithms with improved convergence properties and theoretical guarantees.