Equivariant Covariance Tensors Enhance Geometric Learning Uncertainty.

Geometric deep learning, particularly for tensor-valued predictions, is a critical area, but quantifying uncertainty in such outputs remains a significant challenge. While E(3)-equivariant neural networks excel at providing point estimates that respect rotational symmetry, they often lack robust confidence measures. This work addresses this gap by introducing a novel framework for E(3)-equivariant uncertainty quantification (UQ). The framework focuses on symmetric rank-2 tensor prediction, where the target has six Kelvin-Mandel coordinates, and the full uncertainty is represented by a 6x6 covariance matrix. The key innovation is decomposing the covariance into irreducible representations and mapping from the flat Lie algebra to the curved Symmetric Positive Definite (SPD) manifold via matrix exponentiation. This ensures that the predicted covariances are strictly positive-definite while maintaining exact equivariance. Furthermore, the researchers developed a Log-Euclidean Equivariant Scoring Objective (LE-ESO), a robust surrogate loss function based on the Multivariate Laplace distribution. This objective provides resilience against heavy-tailed errors and facilitates stable optimization. Validation on ModelNet40 inertia tensors and Materials Project dielectric tensors demonstrates that the method achieves competitive performance, delivering physically consistent, symmetry-preserving uncertainty estimates with valuable risk and out-of-distribution sensitivity.