# Signal processing on simplicial complexes

arXiv: https://arxiv.org/abs/2106.07471 · HTML: https://mlsanity.com/paper/2106.07471/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 54 references that are in the map.
- First posted on arXiv 14 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [HodgeNet: Graph Neural Networks for Edge Data](https://mlsanity.com/paper/1912.02354/) (2019)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [Signal Recovery on Graphs: Variation Minimization](https://mlsanity.com/paper/1411.7414/) (2014)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data](https://mlsanity.com/paper/1601.01704/) (2016)
- [Simplicial complexes: higher-order spectral dimension and dynamics](https://mlsanity.com/paper/2001.05934/) (2020)
- [Network Inference from Consensus Dynamics with Unknown Parameters](https://mlsanity.com/paper/1908.01393/) (2019)
- [Reconstruction of Graph Signals through Percolation from Seeding Nodes](https://mlsanity.com/paper/1507.08364/) (2015)

## Built on by (most cited first, 12 of 12)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Molecular topological deep learning for polymer property prediction](https://mlsanity.com/paper/2410.04765/) (2024)
- [Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance](https://mlsanity.com/paper/2410.16935/) (2024)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [Higher-order Cross-structural Embedding Model for Time Series Analysis](https://mlsanity.com/paper/2410.22984/) (2024)
- [Efficient Sparsification of Simplicial Complexes via Local Densities of States](https://mlsanity.com/paper/2502.07558/) (2025)
- [Faster Inference of Cell Complexes from Flows via Matrix Factorization](https://mlsanity.com/paper/2508.21372/) (2025)
- [Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications](https://mlsanity.com/paper/2510.23235/) (2025)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Periodic Topological Deep Learning for Polymer Design and Discovery](https://mlsanity.com/paper/2605.26833/) (2026)
- [TopoEmbedX: A General Framework for Representation Learning on Topological Domains](https://mlsanity.com/paper/2609.37884/) (2026)

## Abstract (arXiv)

Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.