Laplacian Operator

The Laplacian operator, a fundamental tool in mathematics and physics, is being actively researched for its applications in diverse fields like geometry processing and machine learning. Current research focuses on extending its use beyond traditional domains, including adapting it for irregular data structures like point clouds using graph neural networks and developing novel variants for directed and signed graphs through techniques like effective adjacency matrices. These advancements improve the accuracy and efficiency of Laplacian-based algorithms, impacting areas such as image denoising, manifold learning, and spectral embedding, with applications ranging from 3D modeling to signal processing.