symmetry

该论文基于等变度数理论（equivariant degree theory）提出了一种通用的拓扑对称性指数，用于刻画神经网络损失景观中临界轨道的各向同性及其净拓扑贡献，适用于任意网络架构并给出软激活 MLP、Transformer 和等变神经网络的计算示例。

MIT, Boston Children's Hospital and Inria researchers presented E3-Pose, a deep learning model with baked-in left-right symmetry physics for tracking fetal head motion, aiming to produce reliable high-resolution fetal brain MRI for prenatal diagnosis.

In a 1982 interview, Nobel laureate Paul Dirac discusses the role of symmetry in physics, recounts the development of the Dirac equation and the discovery of electron spin, and critiques modern theoretical physics.

The paper studies training dynamics for two-layer neural networks learning orthogonal multi-index models under standard initialization, revealing incremental learning and competitive parameter reallocation, and introduces a symmetry-based finite-width approximation for analysis.

A web-based mandala drawing tool that allows users to create symmetric art by drawing a single stroke, with customizable settings for symmetry, density, and ink types.

This paper investigates the role of parameter symmetry in weight-space perception, showing that symmetry scatter alone accounts for nearly the entire degradation in performance between shared and independently-fitted neural networks. The study uses SIRENs and large-scale experiments to argue that computational advantages may justify weight-space methods over informational equivalence to function access.

This paper introduces SymBuild, a method that treats terminal symmetry as a reusable decision resource for sequential construction tasks, using a transport-refine-certify decomposition to improve anytime verified construction performance.

This paper analyzes reasoning shortcuts in neurosymbolic systems through symmetry and automorphism groups, reveals flaws in a prior framework, proves complexity-theoretic results, and empirically validates the corrected theory on benchmarks like CLE4EVR and rsbench.

A study on how KataGo's Go-playing neural networks internally represent board symmetries, finding that the model learns partially symmetric concepts despite only stochastic 8-fold data augmentation, with one unexpected result.

This paper identifies a fundamental issue in top-k explanations for graph neural networks: automorphisms in input graphs cause non-unique explanations, as the model cannot distinguish symmetric elements. The authors provide a criterion to detect such arbitrariness and verify it using automated reasoning in Lean 4, showing the problem is widespread in molecular datasets.

This paper develops a mathematical framework for analyzing information discarded by machine learning models under Lie group actions, introducing null fibers and stabilizers, with applications to data masking, model fingerprinting, and privacy-preserving computation tested on molecular and image tasks.

This paper proposes incorporating symmetries into affinity kernels for spectral embedding, proving convergence of invariant graph Laplacians on quotient manifolds with improved sample complexity.

This paper shows that the geometric symmetry visible from neural network weights depends on the positional encoding and readout observable, and validates this using MLPs trained on 2D signed distance functions with multiple symmetry groups.

This paper studies data augmentation for Bayesian neural networks trained with variational inference, deriving conditions for exact equivariance and introducing novel symmetrization techniques like orbit expansion to improve symmetry and performance.

This paper presents a homotopy-type-theoretic generalization of neurosymbolic inference that preserves symmetry information and proof multiplicity, showing that this framework recovers classical inference when symmetries are trivial and yields shortcut-aware concept posteriors computable in closed form, with practical improvements on reasoning-shortcut benchmarks.

The paper proposes treating fairness as a symmetry operation in machine learning classifiers, implementing loss-based regularization to enforce invariance under swapping of sensitive attributes while holding merit features fixed. The framework achieves over 90% bias reduction with minimal accuracy loss and requires no causal graph knowledge.

This paper empirically measures the symmetry–data exchange rate predicted by equivariance theory, finding that wrong-group symmetry constraints are actively harmful, augmentation with test-time orbit averaging matches equivariant architectures, and the theoretical |G|-fold sample complexity reduction is only weakly confirmed with wide confidence intervals. The study is explicitly exploratory and not pre-registered.

This exploratory study empirically measures the symmetry–data exchange rate predicted by equivariance theory on controlled C_n-symmetric tasks, finding that wrong-group constraints are actively harmful, augmentation with test-time orbit averaging matches equivariant models exactly, and the empirical exchange rate is broadly consistent with theory but statistically inconclusive. The authors emphasize the study's exploratory nature and call for registered replications.

This paper presents a method to automatically generate local search neighborhoods from constraint specifications using symmetry properties, evaluated on six optimization problems.

Researchers introduce symmetry-compatible optimizers that respect the equivariance structures of neural network parameters, improving training stability and performance over traditional methods like Adam. The approach is validated on various language model architectures including Qwen3-0.6B, Gemma 3 1B, and OLMoE-1B-7B.