Motivated by the need to address the degeneracy of canonical Laplace learning
algorithms in low label rates, we propose to reformulate graph-based
semi-supervised learning as a nonconvex generalization of a \emph{Trust-Region
Subproblem} (TRS). This reformulation is motivated by the well-posedness of
Laplacian eigenvectors in the limit of infinite unlabeled data. To solve this
problem, we first show that a first-order condition implies the solution of a
manifold alignment problem and that solutions to the classical \emph{Orthogonal
Procrustes} problem can be used to efficiently find good classifiers that are
amenable to further refinement. Next, we address the criticality of selecting
supervised samples at low-label rates. We characterize informative samples with
a novel measure of centrality derived from the principal eigenvectors of a
certain submatrix of the graph Laplacian. We demonstrate that our framework
achieves lower classification error compared to recent state-of-the-art and
classical semi-supervised learning methods at extremely low, medium, and high
label rates. Our code is available on github\footnote{anonymized for
submission}.

该研究论文提出了一种基于图的半监督学习的非凸泛化解决方案，通过使用 Laplacian 特征向量解决了低标签率下标准算法退化的问题，并通过选择信息样本实现了较低的分类错误率。

在 Stiefel 流形上的半监督拉普拉斯学习

Semi-Supervised Laplacian Learning on Stiefel Manifolds

Polygonal meshes have become the standard for discretely approximating 3D
shapes, thanks to their efficiency and high flexibility in capturing
non-uniform shapes. This non-uniformity, however, leads to irregularity in the
mesh structure, making tasks like segmentation of 3D meshes particularly
challenging. Semantic segmentation of 3D mesh has been typically addressed
through CNN-based approaches, leading to good accuracy. Recently, transformers
have gained enough momentum both in NLP and computer vision fields, achieving
performance at least on par with CNN models, supporting the long-sought
architecture universalism. Following this trend, we propose a transformer-based
method for semantic segmentation of 3D mesh motivated by a better modeling of
the graph structure of meshes, by means of global attention mechanisms. In
order to address the limitations of standard transformer architectures in
modeling relative positions of non-sequential data, as in the case of 3D
meshes, as well as in capturing the local context, we perform positional
encoding by means the Laplacian eigenvectors of the adjacency matrix, replacing
the traditional sinusoidal positional encodings, and by introducing
clustering-based features into the self-attention and cross-attention
operators. Experimental results, carried out on three sets of the Shape COSEG
Dataset, on the human segmentation dataset proposed in Maron et al., 2017 and
on the ShapeNet benchmark, show how the proposed approach yields
state-of-the-art performance on semantic segmentation of 3D meshes.

提出了一种基于 Transformer 以及 Laplacian 特征编码的多边形网格的语义分割方法，并在 Shape COSEG 数据集、人类分割数据集以及 ShapeNet 基准上实现了最先进的性能。

MeT：用于三维网格语义分割的图变换器

MeT: A Graph Transformer for Semantic Segmentation of 3D Meshes

The lack of anisotropic kernels in graph neural networks (GNNs) strongly
limits their expressiveness, contributing to well-known issues such as
over-smoothing. To overcome this limitation, we propose the first globally
consistent anisotropic kernels for GNNs, allowing for graph convolutions that
are defined according to topologicaly-derived directional flows. First, by
defining a vector field in the graph, we develop a method of applying
directional derivatives and smoothing by projecting node-specific messages into
the field. Then, we propose the use of the Laplacian eigenvectors as such
vector field. We show that the method generalizes CNNs on an $n$-dimensional
grid and is provably more discriminative than standard GNNs regarding the
Weisfeiler-Lehman 1-WL test. We evaluate our method on different standard
benchmarks and see a relative error reduction of 8% on the CIFAR10 graph
dataset and 11% to 32% on the molecular ZINC dataset, and a relative increase
in precision of 1.6% on the MolPCBA dataset. An important outcome of this work
is that it enables graph networks to embed directions in an unsupervised way,
thus allowing a better representation of the anisotropic features in different
physical or biological problems.

本研究提出首个全局一致的 anisotropic kernels，允许 GNN 根据拓扑推导的方向流进行图卷积；方法基于图中定义向量场，通过将节点特定信息投影到向量场中应用方向导数和平滑，采用 Laplacian 特征向量作为向量场；研究揭示该方法在 Weisfeiler-Lehman 1-WL 测试方面比标准 GNN 更具有辨别力，且未监督地嵌入方向，能更好地表示不同物理或生物问题的各向异性特征。