# Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes

arXiv: https://arxiv.org/abs/2109.08728 · HTML: https://mlsanity.com/paper/2109.08728/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2021.

## Built on by (most cited first, 11 of 11)

- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Towards Group Robustness in the presence of Partial Group Labels](https://mlsanity.com/paper/2201.03668/) (2022)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [Graph Classification Gaussian Processes via Hodgelet Spectral Features](https://mlsanity.com/paper/2410.10546/) (2024)
- [Multiscale Hodge Scattering Networks for Data Analysis](https://mlsanity.com/paper/2311.10270/) (2023)
- [Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations](https://mlsanity.com/paper/2505.10877/) (2025)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Hodge-Aware Contrastive Learning](https://mlsanity.com/paper/2309.07364/) (2023)
- [SC-MAD: Mixtures of Higher-order Networks for Data Augmentation](https://mlsanity.com/paper/2309.07453/) (2023)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [HodgePhaseNet: Discrete Hodge Decomposition of Semantic Flows in Transformer Attention](https://mlsanity.com/paper/2610.14971/) (2026)

## Abstract (arXiv)

We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to construct wavelets that respect the Hodge-Helmholtz decomposition. For these Hodgelets, familiar notions of curl-free and divergence-free flows from vector calculus are preserved. We characterize the representational quality of our Hodgelets for edge flows in terms of frame bounds and demonstrate the use of these spectral wavelets for sparse representation of edge flows on real and synthetic data.