Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery

• URL: http://arxiv.org/abs/2605.20440v1
• Date: Tue, 19 May 2026 19:47:40 GMT
• Title: Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery
• Abstract summary: We introduce the $star_G$ tensor algebra, in which any finite group $G$ defines the algebraic multiplication rule.<n>The framework provides capabilities that redesign neural networks (ENNs) structurally cannot: a closed-form per-irreducible equi- decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset.
• Score: 17.340037184664208
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce the $\star_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the $\star_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1} \otimes F_{G_2}$ with no architectural redesign; and (iii)~a 600-line Lean~4 formalization of the $\star_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner--Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A$_1$-dominated, dipole components are T$_1$-dominated, the isotropic polarizability is uniquely insensitive to $l\!=\!1$ as the rank-2-trace decomposition $l\!=\!0 \oplus l\!=\!2$ requires, and the T$_1$/A$_1$ predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules), $\star_G$-SVD with ridge regression provides closed form predictions at $\sim50-90\times$ fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.

Related papers

• $\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition [47.80325814873361]
  Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime.<n>We introduce the $mathbbSL(n)$ space, a representation geometry defined by the simple $det(A)variant=1$ constraint and a left in Schatten-$p$ structure.<n>Despite this minimal construction, $mathbbSL(n)$ exhibits pointwise negative, zero, and flag curvature around a common flagpole.
  arXiv  Detail & Related papers  (2026-09-14T05:52:14Z)
• Observable- and Positional-Encoding-Dependent Symmetry Readout from Neural Network Weights [2.595513192756607]
  Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters.<n>We show that, for positional-encoding-separated neural fields, the symmetry visible from weights is not the true symmetry group itself, but an observable symmetry set determined by the trained parameters.
  arXiv  Detail & Related papers  (2026-07-03T08:43:47Z)
• An Information-Geometric Justification for Composite Coherence in Event-Based Narrative Extraction [0.27074235008521247]
  We study the composite metric $C=sqrtAcdot T$, where $A$ is the angular similarity of document embeddings and $T=1-d_mathrmJS$.<n>Experiments on four corpora, three embedding families, and three topic models are consistent with the framework.
  arXiv  Detail & Related papers  (2026-06-28T00:05:15Z)
• On the Equivariant Learning of the $Q$-tensor Order Parameter [0.7734726150561086]
  We construct and evaluate group-equivariant neural networks for the prediction of the two-dimensional $Q$-tensor order parameter of nematic liquid crystals.<n>We find that the equivariant models consistently achieve lower errors and generalize more robustly to unseen defect configurations.
  arXiv  Detail & Related papers  (2026-05-26T20:54:38Z)
• Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds [61.14405512940818]
  We show that Score-based generative models are trained in high-dimensional ambient spaces.<n>We use a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss Newton.
  arXiv  Detail & Related papers  (2026-05-15T10:20:05Z)
• Anisotropic local law for non-separable sample covariance matrices [10.181748307494608]
  We establish local laws for sample covariance matrices $K = N-1sum_i=1N g_ig_ig_i*$ where the random vectors $g_1, ldots, g_N in Rn$ are independent with common covariance $$.<n>We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model $g = (Xw)$ arising in machine learning, and Gaussian measures with
  arXiv  Detail & Related papers  (2026-02-20T03:28:51Z)
• Robust inverse material design with physical guarantees using the Voigt-Reuss Net [0.0]
  We propose a spectrally normalized surrogate for forward and inverse mechanical homogenization with hard physical guarantees.<n>In 3D linear elasticity on an open dataset of biphasic microstructures, a fully connected Voigt-Reuss net trained on $>!7.5times 105$ FFT-based labels with 236 isotropy-in descriptors.<n>Overall, the Voigt-Reuss net unifies accurate, physically admissible forward prediction with large-batch, constraint-consistent inverse design.
  arXiv  Detail & Related papers  (2025-11-14T15:17:37Z)
• Symmetry-Breaking Descent for Invariant Cost Functionals [0.0]
  We study the problem of reducing a task cost functional $W : Hs(M) to mathbbR$, not assumed continuous or differentiable.<n>We show that symmetry-breaking deformations of the signal can reduce the cost.
  arXiv  Detail & Related papers  (2025-05-19T15:06:31Z)
• Fast, Expressive SE$(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space [15.495593104596399]
  We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs. We develop an efficient equivariant group convolutional network for processing 3D point clouds.
  arXiv  Detail & Related papers  (2023-10-04T17:06:32Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Equivalence Between SE(3) Equivariant Networks via Steerable Kernels and Group Convolution [90.67482899242093]
  A wide range of techniques have been proposed in recent years for designing neural networks for 3D data that are equivariant under rotation and translation of the input. We provide an in-depth analysis of both methods and their equivalence and relate the two constructions to multiview convolutional networks. We also derive new TFN non-linearities from our equivalence principle and test them on practical benchmark datasets.
  arXiv  Detail & Related papers  (2022-11-29T03:42:11Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• Stochastic behavior of outcome of Schur-Weyl duality measurement [45.41082277680607]
  We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. We derive various types of distribution including a kind of central limit when $n$ goes to infinity.
  arXiv  Detail & Related papers  (2021-04-26T15:03:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.