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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, Yangchen Zeng, Kangning Cui, Shuaishuai Cao, Simon Fong

TL;DR

SphUnc, a unified framework combining hyperspherical representation learning with structural causal modeling, is introduced, establishing a geometric-causal foundation for uncertainty-aware reasoning in multi-agent settings with higher-order interactions.

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.

SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry

TL;DR

SphUnc, a unified framework combining hyperspherical representation learning with structural causal modeling, is introduced, establishing a geometric-causal foundation for uncertainty-aware reasoning in multi-agent settings with higher-order interactions.

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.
Paper Structure (51 sections, 7 theorems, 30 equations, 15 figures, 9 tables, 1 algorithm)

This paper contains 51 sections, 7 theorems, 30 equations, 15 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Hyperspherical representations and vMF modeling
  4. Uncertainty decomposition: epistemic and aleatoric perspectives
  5. Higher-order relational models and hypergraphs
  6. Causal discovery and interventional analysis in networks
  7. Uncertainty-aware applications and robustness
  8. Computational geometry and optimization
  9. Summary of gaps and contributions
  10. Methodology
  11. Problem formulation
  12. Motivation for hyperspherical representation
  13. Hyperspherical beliefs and entropy
  14. Uncertainty decomposition and identifiability conditions
  15. Structural causal model over spherical latents and causal uncertainty
  16. ...and 36 more sections

Key Result

Theorem 1

For the vMF family on $\mathbb{S}^{D-1}$ with $D\ge 2$, the hyperspherical entropy $\mathcal{H}_{\mathrm{sph}}(\kappa)$ defined in Eq. eq:H_identity satisfies:

Figures (15)

  • 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.
  • 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.
  • ...and 10 more figures

Theorems & Definitions (14)

  • Theorem 1: Monotonicity and limit values of hyperspherical entropy
  • proof
  • Proposition 1: Identifiability of the uncertainty decomposition
  • proof
  • Lemma 1: Limits of angular attention
  • proof
  • Theorem 2: Finite-sample interventional consistency
  • proof
  • Lemma 2: Approximate angular preservation under random projection
  • proof
  • ...and 4 more