# Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning

arXiv: https://arxiv.org/abs/2412.05132 · HTML: https://mlsanity.com/paper/2412.05132/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 75 references that are in the map.
- First posted on arXiv 6 Dec 2024.

## Builds on (most cited first, 30 of 31)

- [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)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [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)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [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)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Sheaf Hypergraph Networks](https://mlsanity.com/paper/2309.17116/) (2023)
- [From probabilistic graphical models to generalized tensor networks for supervised learning](https://mlsanity.com/paper/1806.05964/) (2018)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [Clique topology reveals intrinsic geometric structure in neural correlations](https://mlsanity.com/paper/1502.06172/) (2015)
- [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)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [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)
- [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)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Weighted simplicial complexes and their representation power of higher-order network data and topology](https://mlsanity.com/paper/2207.04710/) (2022)
- [What can topology tell us about the neural code?](https://mlsanity.com/paper/1605.01905/) (2016)
- [Explosive higher-order Kuramoto dynamics on simplicial complexes](https://mlsanity.com/paper/1912.04405/) (2019)

## Built on by (most cited first, 1 of 1)

- [Beyond holography: the entropic quantum gravity foundations of image processing](https://mlsanity.com/paper/2503.14048/) (2025)

## Abstract (arXiv)

Topological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of Topological Machine Learning, great attention has been devoted to signal processing of such topological signals. Most of the previous topological signal processing algorithms treat node and edge signals separately and work under the hypothesis that the true signal is smooth and/or well approximated by a harmonic eigenvector of the Hodge-Laplacian, which may be violated in practice. Here we propose Dirac-equation signal processing, a framework for efficiently reconstructing true signals on nodes and edges, also if they are not smooth or harmonic, by processing them jointly. The proposed physics-inspired algorithm is based on the spectral properties of the topological Dirac operator. It leverages the mathematical structure of the topological Dirac equation to boost the performance of the signal processing algorithm. We discuss how the relativistic dispersion relation obeyed by the topological Dirac equation can be used to assess the quality of the signal reconstruction. Finally, we demonstrate the improved performance of the algorithm with respect to previous algorithms. Specifically, we show that Dirac-equation signal processing can also be used efficiently if the true signal is a non-trivial linear combination of more than one eigenstate of the Dirac equation, as it generally occurs for real signals.