The paper studies the asymptotic behavior of the solution to the p-biharmonic equation on random geometric graphs as the number of data points goes to infinity. The key insights are:
The p-biharmonic equation on graphs can be interpreted as a natural extension of the graph p-Laplacian from the perspective of hypergraphs.
The continuum limit of the p-biharmonic equation on graphs is shown to be an appropriately weighted p-biharmonic equation with homogeneous Neumann boundary conditions.
The proof relies on establishing uniform Lp estimates for solutions and gradients of nonlocal and graph Poisson equations. The L∞ estimates of solutions are also obtained as a byproduct.
The consistency of the graph Laplacian and the classical Laplacian is analyzed for sufficiently smooth functions with homogeneous Neumann boundary conditions.
The results provide insights into the connection between discrete and continuous higher-order regularization models, which is crucial for applications in machine learning and data processing.
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by Kehan Shi,Ma... at arxiv.org 05-01-2024
https://arxiv.org/pdf/2404.19689.pdfDeeper Inquiries