# Simplicial Convolutional Filters

arXiv: https://arxiv.org/abs/2201.11720 · HTML: https://mlsanity.com/paper/2201.11720/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 58 references that are in the map.
- First posted on arXiv 27 Jan 2022.

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

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [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)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (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)
- [Autoregressive Moving Average Graph Filtering](https://mlsanity.com/paper/1602.04436/) (2016)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)
- [Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes](https://mlsanity.com/paper/2109.08728/) (2021)
- [Go with the Flow? A Large-Scale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory](https://mlsanity.com/paper/2110.09637/) (2021)
- [Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution](https://mlsanity.com/paper/1303.6241/) (2013)
- [Metrics in the Space of High Order Networks](https://mlsanity.com/paper/1507.01994/) (2015)

## Built on by (most cited first, 18 of 18)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [ICML 2023 Topological Deep Learning Challenge : Design and Results](https://mlsanity.com/paper/2309.15188/) (2023)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Learning graphs and simplicial complexes from data](https://mlsanity.com/paper/2312.10545/) (2023)
- [Binarized Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2405.04098/) (2024)
- [Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning](https://mlsanity.com/paper/2409.05135/) (2024)
- [Matched Topological Subspace Detector](https://mlsanity.com/paper/2504.05892/) (2025)
- [Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes](https://mlsanity.com/paper/2505.23160/) (2025)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Adaptive Spatio-temporal Estimation on the Graph Edges via Line Graph Transformation](https://mlsanity.com/paper/2311.00656/) (2023)
- [Topological Neural Networks over the Air](https://mlsanity.com/paper/2502.10070/) (2025)
- [Topological Kalman Filtering on Cell Complexes](https://mlsanity.com/paper/2605.15955/) (2026)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling](https://mlsanity.com/paper/2608.01318/) (2026)
- [Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc](https://mlsanity.com/paper/2608.14556/) (2026)
- [Topological Signal Processing With Unoriented Operators](https://mlsanity.com/paper/2609.25310/) (2026)

## Abstract (arXiv)

We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.