{"slug": "equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued", "title": "Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learning", "summary": "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.", "body_md": "arXiv:2608.24386v1 Announce Type: new\nAbstract: 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.", "url": "https://wpnews.pro/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued", "canonical_source": "https://www.machinebrief.com/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-fo-heoz", "published_at": "2026-08-26 04:00:00+00:00", "updated_at": "2026-08-26 06:14:00.025570+00:00", "lang": "en", "topics": ["machine-learning", "artificial-intelligence"], "entities": ["arXiv", "ModelNet40", "Materials Project"], "alternates": {"html": "https://wpnews.pro/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued", "markdown": "https://wpnews.pro/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued.md", "text": "https://wpnews.pro/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued.txt", "jsonld": "https://wpnews.pro/news/equivariant-covariance-tensors-guaranteed-spd-uncertainty-for-tensor-valued.jsonld"}}