研究超图上的半监督学习，发现有效标签传播的条件并提出新正则化方法。
Analysis of Semi-Supervised Learning on Hypergraphs
- 基于随机几何超图建模多体交互，分析标签传播机制。
- 证明离散解收敛到密度加权p-Laplacian方程的连续解。
- 提出高阶超图学习框架，适合点云等复杂数据建模。
超图天然适用于建模多体交互。本文分析了一类定义在随机几何超图上的变分半监督学习问题，并在大数据极限下建立了渐近一致性。特别地，我们识别出确保问题良定性的缩放范围——避免标签传播退化为常数标签——并证明离散最小化器在连续极限下收敛至密度加权p-Laplacian方程的解。此外，我们提出了高阶超图学习（HOHL），一种基于超图诱导子图拉普拉斯幂次的多尺度正则化方案。针对几何点云，我们分析了HOHL的一种高效多尺度拉普拉斯近似，并证明其收敛至高阶Sobolev型半范数。标准基准上的数值实验验证了该高阶正则化的实际有效性。
原文摘要 · Abstract (English)
Hypergraphs provide a natural framework for modeling multiway interactions. We analyze a class of variational semi-supervised learning problems posed on random geometric hypergraphs and establish asymptotic consistency in the large-data limit. In particular, we identify scaling regimes that ensure well-posedness--yielding nontrivial label propagation rather than collapse to a constant labeling--and show that discrete minimizers converge, in the continuum, to solutions of a density-weighted p-Laplacian equation. We also propose Higher-Order Hypergraph Learning (HOHL), a multiscale regularization scheme based on powers of Laplacians associated with hypergraph-induced subgraphs. For geometric point clouds, we analyze an efficient multiscale Laplacian surrogate for HOHL and prove convergence to a higher-order Sobolev-type seminorm. Numerical experiments on standard benchmarks support the practical utility of the resulting higher-order regularization.
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