发表机构
College of Intelligent Robotics and Advanced Manufacturing, Fudan University; School of Microelectronics, Fudan University; ByteDance; School of Information Science and Engineering, East China University of Science and Technology（复旦大学智能机器人与先进制造学院; 复旦大学微电子学院; 字节跳动; 华东理工大学信息科学与工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对几何深度学习中张量值预测的不确定性量化难题，提出E(3)等变UQ框架，通过分解协方差与矩阵指数法保证正定性，构建LE-ESO损失，在两类数据集上验证了方法的竞争力与不确定性估计的有效性。
AI 中文摘要
张量值预测是几何深度学习的基础，但这类输出的不确定性量化（UQ）仍是未解决的挑战。E(3)等变神经网络在点估计方面表现出色，但缺乏严格的置信度度量。我们聚焦于对称秩2张量预测，其目标具有6个Kelvin-Mandel坐标，完全不确定性由6×6协方差矩阵表示。我们提出了一种E(3)等变UQ框架，用于建模完全预测分布，其中均值和协方差均保持旋转对称性。我们的方法将协方差分解为不可约表示：Sym²(ρ_c) ≅ 2×(l=0) ⊕ 2×(l=2) ⊕ 1×(l=4)。通过矩阵指数法将平坦李代数𝔰𝔶𝔪(6)映射到弯曲的SPD流形，我们严格确保协方差的正定性，同时保持精确等变性。此外，我们构建了对数欧几里得等变评分目标（LE-ESO）——一种基于多元拉普拉斯分布的鲁棒替代损失，对重尾误差具有鲁棒性，且优化稳定。在ModelNet40惯性张量和Materials Project介电张量上的验证表明，我们的方法取得了有竞争力的性能，且提供了物理一致、保持对称性的不确定性估计，具有实用的风险和分布外（OOD）敏感性。
英文摘要
Tensor-valued prediction is fundamental to geometric deep learning, yet uncertainty quantification (UQ) for such outputs remains an open challenge. While E(3)-equivariant neural networks excel at point estimates, they lack rigorous confidence measures. We focus on symmetric rank-2 tensor prediction, where the target has six Kelvin--Mandel coordinates and full uncertainty is represented by a $6\times6$ covariance matrix. We introduce a framework for E(3)-equivariant UQ, modeling the full predictive distribution where both mean and covariance preserve rotational symmetry. Our approach decomposes the covariance into irreducible representations $\mathrm{Sym}^2(ρ_c) \cong 2\times(l=0) \oplus 2\times(l=2) \oplus 1\times(l=4)$. By mapping from the flat Lie algebra $\mathfrak{sym}(6)$ to the curved SPD manifold via matrix exponentiation, we strictly ensure positive-definite covariances while maintaining exact equivariance. Furthermore, we formulate a Log-Euclidean Equivariant Scoring Objective (LE-ESO)---a robust surrogate loss based on the Multivariate Laplace distribution---providing robustness to heavy-tailed errors and stable optimization. Validation on ModelNet40 inertia tensors and Materials Project dielectric tensors demonstrates that our method achieves competitive performance and provides physically consistent, symmetry-preserving uncertainty estimates with useful risk and OOD sensitivity.
CommentsAccepted to ICML 2026
