arXiv Machine Learning July 30, 2026 By Paulina Hoyos, Shashanka Ubaru, Dongsung Huh, Vasileios Kalantzis, Kenneth L. Clarkson, Misha Kilmer, Haim Avron, Lior Horesh

Exact Symmetry as Algebra: A Machine-Verified Tensor Calculus that Enforces Physical Selection Rules

Read the original on arXiv Machine Learning →

arXiv:2605. 20440v2 Announce Type: replace Abstract: Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv Machine Learning.

arXiv Machine Learning

Aug 4

Polyatomic Complexes: A topologically-informed learning representation for atomistic systems

arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.

By Rahul Khorana, Marcus Noack, Jin Qian

llms

arXiv Machine Learning

Jul 9

When Do Geometric Algebra Layers Beat Scalarization? A Controlled Study on SO(3)-Equivariant Vector Laws

arXiv:2607. 06634v1 Announce Type: new Abstract: Compact networks built from Clifford algebra Cl(3,0) primitives are exactly SO(3)-equivariant and learn synthetic 3D vector laws from few samples.

By Fabien Polly

arXiv Machine Learning

Jun 30

Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials

arXiv:2606. 29584v1 Announce Type: cross Abstract: $\mathrm{Cl}(3,0)$ interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly.

By Can Polat, Erchin Serpedin, Mustafa Kurban, Hasan Kurban

benchmarks

arXiv AI

Jun 3

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

arXiv:2605. 18106v3 Announce Type: replace-cross Abstract: A striking geometric disparity has long persisted in the practice of deep learning.

By Tim Tsz-Kit Lau, Weijie Su

llms rag

arXiv Machine Learning

Jul 7

Observable- and Positional-Encoding-Dependent Symmetry Readout from Neural Network Weights

arXiv:2607. 03108v1 Announce Type: new Abstract: Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters.

By Naoya Chiba, Satoshi Sugiyama, Yuki Uranishi

arXiv Machine Learning

Jul 9

Gauge-Invariant Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces

arXiv:2607. 07032v1 Announce Type: new Abstract: Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures.

By Jiaqing Xie, Yuxin Wang

benchmarks

Related stories

Polyatomic Complexes: A topologically-informed learning representation for atomistic systems

When Do Geometric Algebra Layers Beat Scalarization? A Controlled Study on SO(3)-Equivariant Vector Laws

Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

Observable- and Positional-Encoding-Dependent Symmetry Readout from Neural Network Weights

Gauge-Invariant Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces