SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry

Rong Fu; Chunlei Meng; Jinshuo Liu; Dianyu Zhao; Yongtai Liu; Yibo Meng; Xiaowen Ma; Wangyu Wu

cs.LGMar 1, 2026

SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry

Authors: Rong Fu, Chunlei Meng, Jinshuo Liu, Dianyu Zhao, Yongtai Liu, Yibo Meng, Xiaowen Ma, Wangyu Wu,

Organizations: Independent Researcher

Abstract

Reliable decision-making in complex multi-agent systems requires calibrated predictions and interpretable uncertainty. We introduce SphUnc, a unified framework combining hyperspherical representation learning with structural causal modeling. The model maps features to unit hypersphere latents using von Mises-Fisher distributions, decomposing uncertainty into epistemic and aleatoric components through information-geometric fusion. A structural causal model on spherical latents enables directed influence identification and interventional reasoning via sample-based simulation. Empirical evaluations on social and affective benchmarks demonstrate improved accuracy, better calibration, and interpretable causal signals, establishing a geometric-causal foundation for uncertainty-aware reasoning in multi-agent settings with higher-order interactions.

Figures & tables

Figure 1: Overview of the SphUnc framework for hyperspherical uncertainty decomposition and causal identification. The pipeline initiates with Spherical Latent Encoding , mapping multi-agent features onto the unit hypersphere via a Projection-and-Normalization layer. The architecture then bifurcates into two specialized streams: Hyperspherical Uncertainty Quantification , which employs a vMF Concentration Head to compute epistemic entropy and an Aleatoric Head for data noise, integrated by an Information Geometric Fusion module; and Structural Causal Modeling (SCM) , which utilizes Spherical Hypergraph Message Passing with angular attention to identify directed influences. These streams enable both Calibrated Predictive Distributions and Interventional Simulation under the $do$ -calculus. The entire system is optimized via a Composite Learning Objective that balances predictive accuracy, entropy calibration, and causal fidelity.

Dataset	SphUnc vs. Causal-SphHN		SphUnc vs. CI-GNN		SphUnc vs. SphereNet
Dataset	Performance Gain	p-value	Performance Gain	p-value	Performance Gain	p-value
SNARE (F1)	+7.1	0.003*	+10.3	0.001*	+13.5	<0.001*
PHEME (AUC)	+6.6	0.004*	+8.7	0.002*	+11.3	<0.001*
AMIGOS (Acc)	+6.8	0.002*	+11.3	<0.001*	+14.6	<0.001*

Table 1: Paired t-test results (p-values) against SphUnc. Asterisk denotes significance at

p<0.01

Dataset	What Baselines Miss	SphUnc Gain	Example Scenario
SNARE (Offline)	Treats correlations as causal and fails to isolate peer influence from homophily.	+7.1 F1; P@10 increases from 0.62 to 0.78; identifies latent interventions.	SphUnc performs a counterfactual intervention that fixes an agent’s latent state and correctly attributes downstream behavior changes to causal influence rather than shared confounders.
PHEME (Online)	Produces overconfident early predictions on incomplete threads.	+6.6 AUC; ECE reduced from 0.041 to 0.018.	For ambiguous rumor threads, SphUnc reports high epistemic entropy and postpones a confident label until subsequent corroborating posts reduce uncertainty.
AMIGOS (Affect)	Treats all modalities as equally reliable and misclassifies when signals conflict.	+6.8 Accuracy; robust under conflicting cues.	When posture indicates excitement but facial cues are noisy, the aleatoric head down-weights the noisy modality and the fused decision selects the correct affective state.

Table 2: Qualitative case analysis showing where SphUnc improves over baselines.

Model	Dropout 0.0	Dropout 0.2	Dropout 0.4	Dropout 0.6
HyperGCN	63.2	60.1	55.7	48.3
SS-HGNN	66.0	63.2	58.0	51.5
SphereNet	65.0	62.0	57.5	50.8
CI-GNN	68.2	66.1	62.3	56.0
Causal-SphHN	71.4	70.6	68.0	62.1
SphUnc (Ours)	78.5	78.0	77.2	75.8

Table 3: Robustness to node feature dropout on SNARE (F1 score).

Dataset	Nodes	Hyperedges	Classes	Features per Node	Time Points
SNARE (Offline Networks)	540	1,120	3	24	5
PHEME (Online Discourse)	4,830	8,000	4	300	10
AMIGOS (Affect)	40	120	4	128	4
Financial Network (Extended)	5,000	12,500	5	100	20
Collaboration Graph (Extended)	10,000	30,000	10	50	15

Table 4: Dataset statistics.

Component	Value
Spherical Embedding Dimension, $D$	128
vMF Concentration Param, $\kappa$	[1, 200] (learned)
Aleatoric Head Dimension	64
Fusion Head Hidden Layers	2
Learning Rate	$5\times 10^{-4}$
Batch Size	256

Table 5: Training hyperparameters.

Model	SNARE		PHEME		AMIGOS
Model	F1 $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	Acc $\uparrow$	ECE $\downarrow$
HyperGCN	63.2	0.082	71.1	0.088	58.7	0.090
SS-HGNN	66.8	0.074	73.3	0.067	61.9	0.080
SphereNet	65.0	0.069	73.9	0.063	60.5	0.075
CI-GNN	68.2	0.060	76.5	0.059	63.8	0.071
Causal-SphHN	71.4	0.045	78.6	0.041	68.3	0.048

Table 6: Main classification results across datasets. Higher is better for F1, AUC and Accuracy. Lower is better for ECE.

Variant	SNARE		PHEME
Variant	F1	ECE	AUC	ECE
Full SphUnc	78.5	0.022	85.2	0.018
w/o Hyperspherical Rep. (Euclidean)	69.8	0.071	76.1	0.065
w/o Uncertainty Decomposition (Only Aleatoric)	72.3	0.055	79.5	0.048
w/o Uncertainty Decomposition (Only Epistemic)	70.1	0.068	77.8	0.052
w/o Identifiability Constraints (Ad-hoc Fusion)	74.0	0.042	81.3	0.035

Table 7: Ablation study results on SNARE and PHEME. Higher is better for F1 and AUC. Lower is better for ECE.

Model	SNARE (P@10)	PHEME (P@10)	Financial Net (P@10)
HyperGCN	0.35	0.38	0.32
SS-HGNN	0.42	0.46	0.40
CI-GNN	0.45	0.48	0.43
Causal-SphHN	0.62	0.64	0.59
SphUnc (Ours)	0.78	0.81	0.75

Table 8: Causal influence recovery: Precision@10 alignment with expert-labelled influence links.

Model	SNARE	PHEME	AMIGOS	Average
HyperGCN	0.082	0.088	0.090	0.087
SS-HGNN	0.074	0.067	0.080	0.074
SphereNet	0.069	0.063	0.075	0.069
CI-GNN	0.060	0.059	0.071	0.063
Causal-SphHN	0.045	0.041	0.048	0.045
SphUnc (Ours)	0.022	0.018	0.025	0.022

Table 9: Expected Calibration Error (ECE). Lower is better.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Reliability diagram for SNARE: predicted confidence versus observed accuracy for SphUnc and the Causal-SphHN baseline. The dashed diagonal denotes perfect calibration.

Figure 3: Reliability diagram for PHEME: predicted confidence versus observed accuracy.

Figure 4: Reliability diagram for AMIGOS: predicted confidence versus observed accuracy.

Figure 5: Uncertainty decomposition: scatter of epistemic uncertainty (x-axis) versus aleatoric uncertainty (y-axis). Marker size highlights misclassified examples.

Figure 6: Training losses over epochs: total loss and its decomposition into predictive, entropy and causal components.

Figure 7: Training calibration and causal recovery dynamics: Expected Calibration Error (ECE, left axis) and Precision@10 (right axis) over epochs. Legend placed to avoid overlapping the curves.

Figure 8: Relationship between epistemic uncertainty and prediction error. The fitted trendline quantifies how epistemic entropy correlates with empirical mistakes.

Figure 9: Top learned causal edges (directed): green edges indicate matches to expert-annotated influence links; dashed/orange edges indicate model false positives. Node layout uses a force-directed arrangement for readability.

Figure 10: Intervention strength versus interventional entropy $\mathcal{H}_{\mathrm{causal}}$ for example targets. The panel visualizes how uncertainty about downstream outcomes changes with different magnitudes of latent interventions.

Figure 11: Robustness to node feature dropout (SNARE): continuous F1 curves for SphUnc and baselines as dropout rate increases.

Figure 12: Ablation study: grouped bars show component-wise contributions (SNARE F1 and PHEME AUC) when modules or losses are removed.

Figure 13: Histogram of learned vMF concentration parameter $\kappa$ across samples.

Figure 14: Boxplot of learned $\kappa$ values, summarizing central tendency and spread.

Figure 15: Computational cost comparison: training time, inference latency and memory usage for baseline models and SphUnc.

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Quantifying uncertainty in neural network predictions is essential for high-stakes domains such as autonomous driving, healthcare, and manufacturing. While existing approaches often depend on costly sampling or restrictive distributional assumptions, we propose Hyperspherical Confidence Mapping (HCM), a simple yet principled framework for sampling-free and distribution-free uncertainty estimation. HCM decomposes outputs into a magnitude and a normalized direction vector constrained to lie on the unit hypersphere, enabling a novel interpretation of uncertainty as the degree of violation of this geometric constraint. This yields deterministic and interpretable estimates applicable to both regression and classification. Experiments across diverse benchmarks and real-world industrial tasks demonstrate that HCM matches or surpasses ensemble and evidential approaches, with far lower inference cost and stronger confidence-error alignment. Our results highlight the power of geometric structure in uncertainty estimation and position HCM as a versatile alternative to conventional techniques.

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