# Continuum limit of p-biharmonic equations on graphs

arXiv: https://arxiv.org/abs/2404.19689 · HTML: https://mlsanity.com/paper/2404.19689/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 6 of 6)

- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited](https://mlsanity.com/paper/1312.5179/) (2013)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Asymptotic behavior of ℓp-based Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/1603.00564/) (2016)
- [CURE: Curvature Regularization For Missing Data Recovery](https://mlsanity.com/paper/1901.09548/) (2019)
- [Mumford-Shah functionals on graphs and their asymptotics](https://mlsanity.com/paper/1906.09521/) (2019)

## Built on by (most cited first, 1 of 1)

- [Hypergraph p-Laplacian equations for data interpolation and semi-supervised learning](https://mlsanity.com/paper/2411.12601/) (2024)

## Abstract (arXiv)

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 Lp estimates for solutions and gradients of nonlocal and graph Poisson equations. The L^∞ estimates of solutions are also obtained as a byproduct.