Hypergraph p-Laplacian regularization on point clouds for data interpolation

2024-05-02

arXiv.org

AI Summary

This work addresses point cloud interpolation—where no explicit geometric structure is available—by proposing a hypergraph-based $p$-Laplacian regularization framework. To explicitly capture higher-order relationships among points, we construct $varepsilon_n$-ball and $k_n$-nearest-neighbor hypergraphs. We establish, for the first time, the continuous variational consistency of the hypergraph $p$-Laplacian in semi-supervised interpolation, under weaker upper-bound conditions on $varepsilon_n$ and $k_n$ than those required by graph-based models; this mitigates spurious peaks at labeled points. The resulting optimization problem is efficiently solved via the stochastic primal-dual hybrid gradient (SPDHG) algorithm. Experiments demonstrate that our method significantly outperforms graph-based $p$-Laplacian approaches across multiple point cloud interpolation benchmarks. Moreover, interpolations in neighborhoods of labeled points exhibit superior smoothness and stability, empirically validating the effectiveness of hypergraph modeling and higher-order regularization.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.

Problem

• Interpolates point cloud data lacking explicit structure
• Uses hypergraph p-Laplacian for higher-order relations modeling
• Improves regularization with weaker assumptions on parameters

Innovation

• Hypergraph p-Laplacian for point cloud interpolation
• Stochastic primal-dual hybrid gradient optimization
• Improved assumptions on ε_n and k_n bounds