# Topological Signal Processing over Cell Complexes

arXiv: https://arxiv.org/abs/2112.06709 · HTML: https://mlsanity.com/paper/2112.06709/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2021.

## Built on by (most cited first, 24 of 24)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Cell Attention Networks](https://mlsanity.com/paper/2209.08179/) (2022)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module](https://mlsanity.com/paper/2305.16174/) (2023)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [Representing Edge Flows on Graphs via Sparse Cell Complexes](https://mlsanity.com/paper/2309.01632/) (2023)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [P-Tensors: a General Formalism for Constructing Higher Order Message Passing Networks](https://mlsanity.com/paper/2306.10767/) (2023)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [Adaptive Spatio-temporal Estimation on the Graph Edges via Line Graph Transformation](https://mlsanity.com/paper/2311.00656/) (2023)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Faster Inference of Cell Complexes from Flows via Matrix Factorization](https://mlsanity.com/paper/2508.21372/) (2025)
- [Topology of Reasoning: Retrieved Cell Complex-Augmented Generation for Textual Graph Question Answering](https://mlsanity.com/paper/2602.19240/) (2026)
- [No Triangulation Without Representation: Generalization in Topological Deep Learning](https://mlsanity.com/paper/2605.06467/) (2026)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Differentiable Lifting for Topological Neural Networks](https://mlsanity.com/paper/2608.01160/) (2026)
- [Topology-Consistent Task Planning over Cellular Workflow Complexes for LLM-based Agents](https://mlsanity.com/paper/2610.07004/) (2026)

## Abstract (arXiv)

The Topological Signal Processing (TSP) framework has been recently developed to analyze signals defined over simplicial complexes, i.e. topological spaces represented by finite sets of elements that are closed under inclusion of subsets [1]. However, the same inclusion property represents sometimes a too rigid assumption that prevents the application of simplicial complexes to many cases of interest. The goal of this paper is to extend TSP to the analysis of signals defined over cell complexes, which represent a generalization of simplicial complexes, as they are not restricted to satisfy the inclusion property. In particular, the richer topological structure of cell complexes enables them to reveal cycles of any order, as representative of data features. We propose an efficient method to infer the topology of cell complexes from data by showing how their use enables sparser edge signal representations than simplicial-based methods. Furthermore, we show how to design optimal finite impulse response (FIR) filters operating on solenoidal and irrotational signals in order to minimize the approximation error with respect to the desired spectral masks.