Explaining Hyperbolic Neural Networks via Geometry-Aware Relevance Propagation

Ping Xiong; Shanglin Li; Yi Ding; Thomas Schnake; Shinichi Nakajima

cs.LGSep 28, 2026

Explaining Hyperbolic Neural Networks via Geometry-Aware Relevance Propagation

Authors: Ping Xiong, Shanglin Li, Yi Ding, Thomas Schnake, Shinichi Nakajima

Organizations: BIFOLD, Germany · Technische Universität Berlin, Germany · Nanyang Technological University, Singapore · University of Toronto, Canada · Vector Institute for Artificial Intelligence, Canada · RIKEN Center for AIP, Japan

Abstract

Hyperbolic neural networks introduce geometric operations that require explicit treatment in relevance propagation. Equivalent geometric realizations can produce different feature attributions, even when local relevance is conserved. We study this problem through Geometric Representation Invariance (GRI), a specialization of Implementation Invariance, and zero-curvature consistency, which requires identity relevance propagation when a geometric module approaches the identity. We propose LRP-radial-all for origin-centered radial modules, treating geometric scaling as modulation and assigning relevance entirely to the signal branch. The rule conserves relevance, is invariant to equivalent radial factorizations, and satisfies zero-curvature consistency, yielding GRI for a specified Poincaré-Lorentz logarithmic-map construction. In contrast, a conservative LRP-half baseline can violate both consistency criteria. Experiments on hyperbolic MNIST, sEEG, and CIFAR-10 classifiers assess attribution fidelity, qualitative explanations, and runtime. LRP-radial-all achieves competitive attribution fidelity across datasets with runtime comparable to Gradient $\times$ Input and substantially lower than Integrated Gradients. These findings motivate geometry-aware propagation rules that distinguish relevance conservation from consistency across equivalent computations.

Figures & tables

Figure 1: Geometric Representation Invariance (GRI) requires explanations at the same input after equivalent internal coordinate realizations to be the same.

Figure 2: Top: factorization of

\log

and

\exp

map with origin point in Poincaré space. Bottom: our proposed radial-all rule and half rule for LRP applied to such layer.

	MNIST		sEEG		CIFAR-10
Method	$F^{+}\uparrow$	$F^{-}\downarrow$	$F^{+}\uparrow$	$F^{-}\downarrow$	$F^{+}\uparrow$	$F^{-}\downarrow$
LRP ${}_{\text{radial-all}}$	$\mathbf{0.670}_{\pm 0.005}$	$\mathbf{-0.003}_{\pm 0.005}$	$\mathbf{0.313}_{\pm 0.096}$	$-0.111_{\pm 0.017}$	$0.637_{\pm 0.020}$	$\mathbf{0.277}_{\pm 0.019}$
LRP ${}_{\text{half}}$	$0.562_{\pm 0.011}$	$0.063_{\pm 0.008}$	$0.310_{\pm 0.095}$	$-0.106_{\pm 0.017}$	$0.588_{\pm 0.022}$	$0.505_{\pm 0.022}$
Gradient $\times$ Input	$0.410_{\pm 0.017}$	$0.202_{\pm 0.014}$	$0.303_{\pm 0.096}$	$\mathbf{-0.113}_{\pm 0.017}$	$\mathbf{0.703}_{\pm 0.020}$	$0.547_{\pm 0.023}$
Integrated Grad.	$0.664_{\pm 0.005}$	$0.000_{\pm 0.005}$	$0.309_{\pm 0.096}$	$\mathbf{-0.113}_{\pm 0.017}$	$0.683_{\pm 0.020}$	$0.495_{\pm 0.022}$
Patch Occlusion	$0.504_{\pm 0.013}$	$0.067_{\pm 0.007}$	-	-	-	-

Table 1: Sparsity-fidelity AUC on MNIST, sEEG and CIFAR-10.

F^{+}

and

F^{-}

denote Fidelity + and Fidelity - AUC. Subscripts report

\pm h

, where

h=\max(\mu-L,U-\mu)

for the 95% bootstrap interval

[L,U]

. Bold indicates the best mean within each dataset and metric.

Dataset	LRP-radial-all	Gradient $\times$ Input	Occlusion	Integrated Gradients
MNIST (ms/image)	0.505	0.960	12.676	41.511
sEEG (ms/sample)	5.236	3.813	-	94.431
CIFAR-10 (ms/image)	15.688	12.730	-	66.375

Table 2: Runtime comparison of explanation methods. We average the runtime on 3 runs.

Panel of Figure 4 — Figure 4: Averaged signals and relevance heatmaps for class 0 (low working-memory load) and class 1 (high working-memory load). Boxes highlight localized relevance, distributed multichannel relevance, stronger late-window relevance, and class-dependent attribution differences. Each time series stands for a sensor. Time range is 3-second with 3000 points. Red and blue indicate positive and negative relevance for the explained class score, respectively. The full figure is in Figure 11 .

Figure 5: Relevance of sEEG prediction by class in frequency space. The relevance score is averaged on all trials of subject 2 in the dataset. We grouped the frequency-domain relevance into conventional electrophysiological frequency bands: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz), and gamma (30–70 Hz).

Figure 6: Qualitative attribution comparison on a correctly classified CIFAR-10 image of ship. Columns show the input, heatmaps for LRP-radial-all, LRP-half, Gradient $\times$ Input, and Integrated Gradients. RGB-channel attributions are summed, with red and blue indicating positive and negative values, respectively. More examples in Figure 10 .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Redistribution of scalar-branch relevance from

\alpha(\|\bm{x}\|_{2})

\bm{x}

through the squared norm.

Rule	Path	$R_{a_{1}}$	$R_{a_{2}}$	$R_{a_{3}}$
LRP-radial-all	Direct	0.210205	0	0
	Via Lorentz	0.210205	0	0
LRP-half	Direct	0.135132	0.067566	0.007507
	Via Lorentz	0.097595	0.101349	0.011261

Appendix

Table 3: Relevance at the common original input

\bm{a}=\bm{x}_{P}

for the two equivalent logarithmic-map paths. Values are rounded to six decimal places.

Figure 8: Temporal perturbation validation of LRP for low-load (top) and high-load (bottom) trials. Left: window relevance and the five highest-ranked windows. Right: mean target probability drops under LRP-guided versus random perturbation. Shading indicates the 2.5th-97.5th percentiles across 20 random selections.

Figure 9: Fidelity-Sparsity curve for MNIST experiment with 95% confidence interval. For Fidelity+ higher is better and for Fidelity- lower is better. Lower sparsity corresponds to a larger selected pixel set, which is removed for Fidelity+ and retained for Fidelity-.

Figure 10: Qualitative attribution comparison on four CIFAR-10 examples. Columns show the input, heatmaps for LRP-radial-all, LRP-half, Gradient $\times$ Input, and Integrated Gradients. RGB-channel attributions are summed, with red and blue indicating positive and negative values, respectively.

Panel of Figure 11 — Figure 11: Averaged signals and relevance heatmaps for class 0 (low working-memory load) and class 1 (high working-memory load). Boxes highlight localized relevance, distributed multichannel relevance, stronger late-window relevance, and class-dependent attribution differences. Each time series stands for a sensor. Time range from left to right is 3-second with 3000 points. Red and blue indicate positive and negative relevance for the explained class score, respectively.

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Abstract

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