用可学习的霍奇矩阵构建网格Transformer，无需昂贵的谱计算。
HodgeFormer: Transformers for Learnable Operators on Triangular Meshes through Data-Driven Hodge Matrices
- 基于离散外微分构造霍奇拉普拉斯算子，用注意力机制直接学习霍奇矩阵。
- 在网格分割与分类任务中性能接近传统方法，但速度更快。
- 适合需要高效几何深度学习的科研与工业应用。
当前用于网格形状分析的主流Transformer架构依赖于需高成本特征值分解的谱特征进行注意力计算。这些方法通过拉普拉斯矩阵或热核签名生成位置嵌入，并与输入特征拼接。本文受离散外微分中霍奇拉普拉斯算子构造启发（$L := \star_0^{-1} d_0^T \star_1 d_0$），提出新型Transformer层：利用多头注意力机制逼近霍奇矩阵$\star_0$、$\star_1$、$\star_2$，并学习作用于顶点、边、面的离散算子族。该方法实现计算高效，直接学习框架下在网格分割与分类任务中表现相当，同时避免了耗时的特征值分解与复杂预处理。
原文摘要 · Abstract (English)
Currently, prominent Transformer architectures applied on graphs and meshes for shape analysis tasks employ traditional attention layers that heavily utilize spectral features requiring costly eigenvalue decomposition-based methods. To encode the mesh structure, these methods derive positional embeddings, that heavily rely on eigenvalue decomposition based operations, e.g. on the Laplacian matrix, or on heat-kernel signatures, which are then concatenated to the input features. This paper proposes a novel approach inspired by the explicit construction of the Hodge Laplacian operator in Discrete Exterior Calculus as a product of discrete Hodge operators and exterior derivatives, i.e. $(L := \star_0^{-1} d_0^T \star_1 d_0)$. We adjust the Transformer architecture in a novel deep learning layer that utilizes the multi-head attention mechanism to approximate Hodge matrices $\star_0$, $\star_1$ and $\star_2$ and learn families of discrete operators $L$ that act on mesh vertices, edges and faces. Our approach results in a computationally-efficient architecture that achieves comparable performance in mesh segmentation and classification tasks, through a direct learning framework, while eliminating the need for costly eigenvalue decomposition operations or complex preprocessing operations.
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