Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learning

Researchers introduced a framework for E(3)-equivariant uncertainty quantification in tensor-valued geometric learning, modeling full predictive distributions with guaranteed positive-definite covariances via matrix exponentiation. The method, validated on ModelNet40 inertia tensors and Materials Project dielectric tensors, achieves competitive performance and provides symmetry-preserving uncertainty estimates with useful risk and out-of-distribution sensitivity.

arXiv:2608.24386v1 Announce Type: new Abstract: 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 \rho 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.