# HodgeFormer: Transformers for Learnable Operators on Triangular Meshes through Data-Driven Hodge Matrices

arXiv: https://arxiv.org/abs/2509.01839 · HTML: https://mlsanity.com/paper/2509.01839/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 43 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Transformers in Vision: A Survey](https://mlsanity.com/paper/2101.01169/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Efficient Content-Based Sparse Attention with Routing Transformers](https://mlsanity.com/paper/2003.05997/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Mesh Graphormer](https://mlsanity.com/paper/2104.00272/) (2021)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [DiffusionNet: Discretization Agnostic Learning on Surfaces](https://mlsanity.com/paper/2012.00888/) (2020)
- [MeshCNN: A Network with an Edge](https://mlsanity.com/paper/1809.05910/) (2018)
- [Subdivision-Based Mesh Convolution Networks](https://mlsanity.com/paper/2106.02285/) (2021)
- [MeshFormer: High-Quality Mesh Generation with 3D-Guided Reconstruction Model](https://mlsanity.com/paper/2408.10198/) (2024)
- [Primal-Dual Mesh Convolutional Neural Networks](https://mlsanity.com/paper/2010.12455/) (2020)
- [DualConvMesh-Net: Joint Geodesic and Euclidean Convolutions on 3D Meshes](https://mlsanity.com/paper/2004.01002/) (2020)
- [MeshWalker: Deep Mesh Understanding by Random Walks](https://mlsanity.com/paper/2006.05353/) (2020)
- [MeshMAE: Masked Autoencoders for 3D Mesh Data Analysis](https://mlsanity.com/paper/2207.10228/) (2022)
- [Surface Networks](https://mlsanity.com/paper/1705.10819/) (2017)
- [HodgeNet: Learning Spectral Geometry on Triangle Meshes](https://mlsanity.com/paper/2104.12826/) (2021)
- [Laplacian2Mesh: Laplacian-Based Mesh Understanding](https://mlsanity.com/paper/2202.00307/) (2022)
- [E(3)-Equivariant Mesh Neural Networks](https://mlsanity.com/paper/2402.04821/) (2024)
- [MeT: A Graph Transformer for Semantic Segmentation of 3D Meshes](https://mlsanity.com/paper/2307.01115/) (2023)

## Built on by (most cited first, 3 of 3)

- [Intrinsic and Triangulation-Agnostic Attention: A Simple and Powerful Approach for Learning on Meshes](https://mlsanity.com/paper/2607.24954/) (2026)
- [Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc](https://mlsanity.com/paper/2608.14556/) (2026)
- [Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes](https://mlsanity.com/paper/2610.20585/) (2026)

## Abstract (arXiv)

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 := star0-1 d0T star1 d0). We adjust the Transformer architecture in a novel deep learning layer that utilizes the multi-head attention mechanism to approximate Hodge matrices star0, star1 and star2 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.