Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning

TL;DR

This work targets the processing of topological signals defined on networks, where traditional Hodge-Laplacian approaches assume smooth or harmonic signals and separate treatment of node and edge signals. It introduces DESP, a physics-inspired framework based on the Topological Dirac equation that jointly processes node and edge signals by learning a mass parameter and an energy $E$, enabling efficient reconstruction even when signals are non-harmonic or structured as a single Dirac eigenstate; its performance is further enhanced by IDESP, which iterates DESP to handle linear combinations of eigenstates. The approach unifies DSP and LSP as limiting cases and leverages a relativistic dispersion relation to guide parameter learning, achieving superior reconstruction on synthetic networks and real data (e.g., drifter trajectories). These results suggest broad potential for Dirac-based signal processing in topological machine learning, offering improved robustness to scale differences between node and edge signals and avenues for multi-layer or multiplex networks.

Abstract

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.

Figures (7)

• Figure 1: The visualization of the eigenstates of the topological Dirac equation associated with the value of the mass $m=1.5$ on the Network Geometry with Flavor model (NGF) (panel (a)) and on a real fungi network (panel (b)). The eigenstates $E=m=-1.5$ and $E=m=1.5$ are the harmonic eigenstates which are non-zero only on edges ($E=-m=-1.5$) or only on nodes ($E=m=1.5$). The eigenstates with energy $E>m=1.5$ are non-harmonic, they involve non trivially both node and edge signals, and display characteristic localized patterns. These latter signals are typical examples of signal that can be reconstructed with the DESP. The NGF network in panel (a) is a sample of a two dimensional NGF model with parameters $\beta=0$ and flavor $s=-1$$N_0=20$ nodes and $N_1=37$ edges. This model is defined in Refs. bianconi2016networkbianconi2017emergent and the code for generate network in this model is available at the repository gin_repository. The fungi network in panel (b) is the Pp_M_Tokyo_U_N_26h_1.mat, of $N_0 = 411$ nodes, and $N_1 = 645$ edges from Ref. lee2017mesoscale available at the repository benson_repository.
• Figure 2: We consider a true signal given by an eigenstate of the Topological Dirac equation. For any fixed value of $m$ the iterative nature of the DESP algorithm allows to decrease the true error $\Delta \bm\psi$ with time $t$ (panel (a)) and to best approximate the energy of the signal with time $t$ so that if $m$ is the true mass $m=m_{\textrm{true}}$ then the estimated energy $\hat{E}_{t,m}$ converges to the true energy $E_{\textrm{true}}$ (panel (b)). As the algorithm sweeps over different values possible value of the mass $m$, the true value of the mass and the true value of the energy are reliably estimated under very general conditions on the noise level (panels (c) and (d)). Here the DESP convergence to the true mass and energy parameters is demonstrated on the NGF network shown in Figure 1. The true value of the mass is $m_{\textrm{true}}=1.5$ and the true value of the energy is $E_{\textrm{true}}=-3.19$. The noise is generated using a value of the $\alpha$ parameter given by $\alpha = 0.3$, while the loss $\mathcal{L}$ used to detect both the energy and mass has filtration parameter $\tau = 10.$
• Figure 3: The loss $\mathcal{L}_m$, and the RDRE $\mathcal{S}_m$, are plotted versus $m$ when the true signal is aligned to a single eigenstate of the Topological Dirac equation of the NGF (panels (a)-(b)) and the fungi network (panels (d)-(e)) considered in Figure 1. The minimization of the loss $\mathcal{L}_m$ and of the RDRE $\mathcal{S}_m$ lead to different estimated values of the mass (panels (c), (f)). However, the error $\Delta \bm\psi$ corresponding to these two methods to infer the true mass remains small under very general conditions on the noise level (panels (c), (f)). The Topological Dirac equation eigenstates have true parameters $E_\textrm{true}=-3.27$ and $m_{\textrm{true}}=1.5$ for the NGF (panels (a)-(c)) and $E_\textrm{true}=2.57$ and $m_{\textrm{true}}=1.5$ for the fungi network (panels (d)-(f)). The noise level is $\alpha = 0.3$ and $\tau = 10$ for both cases.
• Figure 4: The DESP includes both the LSP and the DSP as subcases, and in general, can outperform both LSP and DSP. In order to compare the methods, we consider first a true signal given by an eigenstate of the Topological Dirac equation with $m=0$ and tunable value of the energy $E$ (indicating in this case the eigenvalue of the Dirac operator where $E=0$). By assuming that the value of the mass $m=0$ is known, DESP reduces to DSP that outperforms LSP (panel (a)) if the signal deviates from an almost harmonic signal (larger values of $|E|$). Indeed the error $\Delta \bm\psi$ of the reconstructed signal is much lower for the DSP than for the LSP for larger values of the energy $E$. Secondly, we consider a true signal given by an eigenstate of the Topological Dirac equation with a tunable value of the mass $m$ and random value of the energy $E$. We show that DESP outperforms DSP by learning the true value of the mass, and the improvement in the error level $\Delta \bm\psi$ is more significant as the absolute value of the mass $m$ becomes larger (panel (b)). Here the results are obtained by considering $100$ noisy signals (the amplitude of the shaded regions indicates standard deviations) on the NGF network shown in Figure 1 with noise level $\alpha = 0.3$. The DESP uses the loss function $\mathcal{L}$ with parameter $\tau =10$ to infer the true mass.
• Figure 5: DESP can jointly process signals on nodes and edges and allow us to extract relevant information across topological signals of different dimensions. Here we plot the error $\Delta \bm\chi=\|\hat{\bm\chi}-\bm\chi\|/\|\bm\chi\|$, where $\hat{\chi}$ is the reconstructed node signal, and $\bm\chi$ is the true node signal as a function of the noise level $\alpha_2$ associated to the edge signal. We show that the error $\Delta \bm\chi=\|\hat{\bm\chi} \|$ made on the reconstruction of the node signal decreases as $\alpha_2$ is lowered, when the noise level on the edge signal is kept equal to $\alpha_2=0.5$ (panel(a)). Similarly, we show the error $\Delta \bm\phi=\|\hat{\bm\phi}-\bm\phi\|/\|\bm\phi\|$ where $\hat{\phi}$ is the reconstructed edge signal, and $\bm\phi$ is the true edge signal as a function of the noise level $\alpha_1$ on the node signal. Also in this case we show that the error $\Delta \bm\phi$ made on the reconstruction of the edge signal decreases as the noise level $\alpha_1$ associated with the node signal is lowered, when the noise level on the edge signal is kept equal to $\alpha_2=0.5$ (panel (b)). The results suggest an improvement in performance when the noise level on either nodes or links is independently reduced. The shaded area refers to the standard deviation error of DESP calculated over $200$ noisy signals of the NGF network shown in Figure 1. In both panels, true signal is an eigenstate of the Topological Dirac equation with true mass $m=1.5$ and true energy $E = -3.31.$