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

arXiv: https://arxiv.org/abs/2608.24386 · HTML: https://mlsanity.com/paper/2608.24386/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 34 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Graph neural networks for materials science and chemistry](https://mlsanity.com/paper/2208.09481/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Better Uncertainty Calibration via Proper Scores for Classification and Beyond](https://mlsanity.com/paper/2203.07835/) (2022)
- [Open Materials 2024 (OMat24) Inorganic Materials Dataset and Models](https://mlsanity.com/paper/2410.12771/) (2024)
- [Tractable Function-Space Variational Inference in Bayesian Neural Networks](https://mlsanity.com/paper/2312.17199/) (2023)
- [Emergent Equivariance in Deep Ensembles](https://mlsanity.com/paper/2403.03103/) (2024)
- [DenseGNN: universal and scalable deeper graph neural networks for high-performance property prediction in crystals and molecules](https://mlsanity.com/paper/2501.03278/) (2025)
- [Multi-Scale Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2302.03580/) (2023)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [A Structured Review of Literature on Uncertainty in Machine Learning & Deep Learning](https://mlsanity.com/paper/2406.00332/) (2024)
- [BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework](https://mlsanity.com/paper/2410.16420/) (2024)
- [High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces](https://mlsanity.com/paper/2412.18263/) (2024)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Neural Network Layers for Prediction of Positive Definite Elastic Stiffness Tensors](https://mlsanity.com/paper/2203.13938/) (2022)
- [Dielectric Tensor Prediction for Inorganic Materials Using Latent Information from Preferred Potential](https://mlsanity.com/paper/2405.09052/) (2024)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)
- [Revisiting the Canonicalization for Fast and Accurate Crystal Tensor Property Prediction](https://mlsanity.com/paper/2410.02372/) (2024)
- [High Accuracy Uncertainty-Aware Interatomic Force Modeling with Equivariant Bayesian Neural Networks](https://mlsanity.com/paper/2304.03694/) (2023)
- [On Uncertainty Calibration for Equivariant Functions](https://mlsanity.com/paper/2510.21691/) (2025)

## Abstract (arXiv)

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 6×6 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 Sym2(ρc) cong 2×(l=0) oplus 2×(l=2) oplus 1×(l=4). By mapping from the flat Lie algebra mathfraksym(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.