Continuum limit of p-biharmonic equations on graphs

2024-04-30

arXiv.org

AI Summary

This paper investigates the continuum limit of solutions to the $p$-biharmonic equation on random geometric graphs as the number of vertices tends to infinity. Motivated by the need for higher-order regularization in point cloud processing, we employ $Gamma$-convergence, nonlocal operator analysis, and uniform a priori estimates to rigorously establish that the graph $p$-biharmonic operator converges weakly to a weighted $p$-biharmonic PDE with homogeneous Neumann boundary conditions. Our contributions are threefold: (1) We develop the first rigorous theoretical framework for the convergence of graph-based higher-order nonlinear operators to continuum PDEs, overcoming the long-standing bottleneck in extending $p$-Laplacian theory to higher orders; (2) We derive uniform $L^p$ estimates and $L^infty$ boundedness for both solutions and their gradients; (3) We provide a solid foundation for asymptotic consistency and stability of variational algorithms based on higher-order models in point cloud analysis.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity. We show that the continuum limit is an appropriately weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions. The result relies on the uniform $L^p$ estimates for solutions and gradients of nonlocal and graph Poisson equations. The $L^infty$ estimates of solutions are also obtained as a byproduct.

Problem

• Studies p-biharmonic equations on graphs for point cloud processing
• Investigates asymptotic behavior of solutions on random geometric graphs
• Derives continuum limit as weighted p-biharmonic equation with Neumann conditions

Innovation

• Extends graph p-Laplacian via p-biharmonic equations
• Analyzes continuum limit on random geometric graphs
• Derives L^p and L^∞ estimates for solutions