# Dirac signal processing of higher-order topological signals

arXiv: https://arxiv.org/abs/2301.10137 · HTML: https://mlsanity.com/paper/2301.10137/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 70 references that are in the map.
- First posted on arXiv 12 Jan 2023.

## Builds on (most cited first, 30 of 38)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [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)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Simplicial Attention Networks](https://mlsanity.com/paper/2204.09455/) (2022)
- [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 Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [The why, how, and when of representations for complex systems](https://mlsanity.com/paper/2006.02870/) (2020)
- [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)
- [Unveiling the higher-order organization of multivariate time series](https://mlsanity.com/paper/2203.10702/) (2022)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2106.07471/) (2021)
- [The topological Dirac equation of networks and simplicial complexes](https://mlsanity.com/paper/2106.02929/) (2021)
- [Dynamics on higher-order networks: A review](https://mlsanity.com/paper/2203.06601/) (2022)
- [Simplicial complexes: higher-order spectral dimension and dynamics](https://mlsanity.com/paper/2001.05934/) (2020)
- [Edges in Brain Networks: Contributions to Models of Structure and Function](https://mlsanity.com/paper/2105.07069/) (2021)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Weighted simplicial complexes and their representation power of higher-order network data and topology](https://mlsanity.com/paper/2207.04710/) (2022)

## Built on by (most cited first, 9 of 9)

- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [On the Sheafification of Higher-Order Message Passing](https://mlsanity.com/paper/2509.23020/) (2025)
- [Sheaf-Laplacian Obstruction and Projection Hardness for Cross-Modal Compatibility on a Modality-Independent Site](https://mlsanity.com/paper/2604.07632/) (2026)
- [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)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)
- [Quantum Topological Data Encoding](https://mlsanity.com/paper/2607.13847/) (2026)

## Abstract (arXiv)

Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.