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Tensor learning with orthogonal, Lorentz, and symplectic symmetries

Wilson G. Gregory, Josué Tonelli-Cueto, Nicholas F. Marshall, Andrew S. Lee, Soledad Villar

TL;DR

This work develops a generic, mathematically principled framework for learning tensor-valued maps that are equivariant under diagonal actions of classical groups, including $O(d)$, the Lorentz group $O(s,d-s)$, and the symplectic group $Sp(d)$. By parameterizing equivariant tensor maps through contractions with $G$-isotropic tensors and, in the vector-input case, via invariant inner products, the authors avoid Clebsch–Gordan coefficients and achieve universal expressivity for tensor polynomials and entire functions. They prove a general theorem characterizing all $G$-equivariant tensor polynomials and extend it to indefinite and symplectic groups, providing concrete corollaries for vector-to-tensor mappings and symmetric-matrix cases. The framework is validated across materials science, time-series analysis, and sparse-vector estimation, where the equivariant models consistently outperform non-equivariant baselines and, in some settings, SoS-based methods. The approach offers a scalable, principled pathway to incorporate symmetry into ML models for physics-informed and geometry-aware tensor problems, with reproducible code and datasets.

Abstract

Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.

Tensor learning with orthogonal, Lorentz, and symplectic symmetries

TL;DR

This work develops a generic, mathematically principled framework for learning tensor-valued maps that are equivariant under diagonal actions of classical groups, including , the Lorentz group , and the symplectic group . By parameterizing equivariant tensor maps through contractions with -isotropic tensors and, in the vector-input case, via invariant inner products, the authors avoid Clebsch–Gordan coefficients and achieve universal expressivity for tensor polynomials and entire functions. They prove a general theorem characterizing all -equivariant tensor polynomials and extend it to indefinite and symplectic groups, providing concrete corollaries for vector-to-tensor mappings and symmetric-matrix cases. The framework is validated across materials science, time-series analysis, and sparse-vector estimation, where the equivariant models consistently outperform non-equivariant baselines and, in some settings, SoS-based methods. The approach offers a scalable, principled pathway to incorporate symmetry into ML models for physics-informed and geometry-aware tensor problems, with reproducible code and datasets.

Abstract

Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.
Paper Structure (37 sections, 27 theorems, 172 equations, 1 figure, 7 tables)

This paper contains 37 sections, 27 theorems, 172 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. O(d)-equivariant polynomial functions
  4. Generalizations to other groups
  5. Numerical Experiments
  6. Discussion
  7. Acknowledgements
  8. Reproducibility
  9. Basic properties of O(d) actions on tensors
  10. Proof of Theorem \ref{['thm:equiv_polynomial']}
  11. Proof of Corollary \ref{['cor:vector_to_tensors']}
  12. Smaller parameterization of O(d)-isotropic tensors
  13. Example of Theorem \ref{['thm:equiv_polynomial']}
  14. Proof of Corollary \ref{['cor:symmetric_matrices']}
  15. Generalization to other linear algebraic groups
  16. ...and 22 more sections

Key Result

Theorem 1

Let $f: \prod_{i=1}^n \mathcal{T}_{k_i}\qty(\mathbb{R}^d, p_i) \to \mathcal{T}_{k'}\qty(\mathbb{R}^d, p')$ be an $\mathop{O}(d)$-equivariant polynomial function of degree at most $R$. Then we may write $f$ as follows: where $c_{\ell_1,\ldots,\ell_r}$ is an $\mathop{O}(d)$-isotropic ${(k_{\ell_1,\ldots,\ell_r} + k')}_{(p_{\ell_1,\ldots,\ell_r}\,p')}$-tensor with order and parity chosen to be consi

Figures (1)

  • Figure 1: Illustration of the method from Corollary \ref{['cor:vector_to_tensors']} with 4 input vectors in $\mathbb R^3$ and a ${2}_{(+)}$-tensor output. The tensor product of inputs includes all 16 possible tensor products of ordered pairs of input vectors, plus the isotropic Kronecker delta, labeled $id$. The coefficients $q_{t,\sigma,J}$ shown here use $\sigma=()$, the identity permutation in $S_{k'}$.

Theorems & Definitions (83)

  • Definition 1: (${k}_{(p)}$-tensors)
  • Definition 2: (Einstein summation notation)
  • Definition 3: (Outer product of tensors)
  • Definition 4: ($k$-contraction)
  • Definition 5: (Permutations of tensor indices)
  • Definition 6: (Invariant and equivariant functions)
  • Definition 7: (Isotropic tensors)
  • Definition 8: (Kronecker delta)
  • Definition 9: (Levi-Civita symbol)
  • Theorem 1
  • ...and 73 more