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.

探索了超图结构在不包含显式结构信息的点云数据插值中的优势，并证明了超图 $p$-Laplacian 正则化与连续型 $p$-Laplacian 正则化的变分一致性，利用随机原始 - 对偶混合梯度算法解决了凸但不可微的大规模优化问题，并通过数值实验验证了超图 $p$-Laplacian 正则化相对于图 $p$-Laplacian 正则化在防止标记点处出现尖峰的效果更好。

点云数据插值上的超图 p-Laplace 正则化

Hypergraph $p$-Laplacian regularization on point clouds for data  interpolation

$p$-Laplacian regularization, rooted in graph and image signal processing,
introduces a parameter $p$ to control the regularization effect on these data.
Smaller values of $p$ promote sparsity and interpretability, while larger
values encourage smoother solutions. In this paper, we first show that the
self-attention mechanism obtains the minimal Laplacian regularization ($p=2$)
and encourages the smoothness in the architecture. However, the smoothness is
not suitable for the heterophilic structure of self-attention in transformers
where attention weights between tokens that are in close proximity and
non-close ones are assigned indistinguishably. From that insight, we then
propose a novel class of transformers, namely the $p$-Laplacian Transformer
(p-LaT), which leverages $p$-Laplacian regularization framework to harness the
heterophilic features within self-attention layers. In particular, low $p$
values will effectively assign higher attention weights to tokens that are in
close proximity to the current token being processed. We empirically
demonstrate the advantages of p-LaT over the baseline transformers on a wide
range of benchmark datasets.

该研究探讨了 $p$-Laplacian 正则化在 transformers 中的应用，通过引入参数 $p$ 控制数据的正则化效果，利用自注意机制实现架构的平滑度，进而提出了一种新型的 transformer 模型 $p$-Laplacian Transformer (p-LaT)，在一系列基准数据集上实验证明其优势。