mlsanityData to 14 Oct 2026

On the Symmetries of Deep Learning Models and their Internal Representations

arXiv 2205.14258 ↗First posted 27 May 2022cs.LG cs.AIcited by 64

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Abstract (arXiv)

Symmetry is a fundamental tool in the exploration of a broad range of complex systems. In machine learning symmetry has been explored in both models and data. In this paper we seek to connect the symmetries arising from the architecture of a family of models with the symmetries of that family's internal representation of data. We do this by calculating a set of fundamental symmetry groups, which we call the intertwiner groups of the model. We connect intertwiner groups to a model's internal representations of data through a range of experiments that probe similarities between hidden states across models with the same architecture. Our work suggests that the symmetries of a network are propagated into the symmetries in that network's representation of data, providing us with a better understanding of how architecture affects the learning and prediction process. Finally, we speculate that for ReLU networks, the intertwiner groups may provide a justification for the common practice of concentrating model interpretability exploration on the activation basis in hidden layers rather than arbitrary linear combinations thereof.

Key facts. Cited by 64 papers in the map (data through 14 Oct 2026). Builds on 38 of its 70 references that are in the map. First posted on arXiv 27 May 2022.

Builds on

38 of 70 references.

References · 38

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 7,005
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  11. 2017Network Dissection: Quantifying Interpretability of Deep Visual Representationscited by 708
  12. 2021Do Vision Transformers See Like Convolutional Neural Networks?cited by 675
26 more
  1. 2015Understanding Neural Networks Through Deep Visualizationcited by 334
  2. 2021The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networkscited by 291
  3. 2021Escaping the Big Data Paradigm with Compact Transformerscited by 264
  4. 2020Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depthcited by 245
  5. 2014Understanding image representations by measuring their equivariance and equivalencecited by 243
  6. 2015Convergent Learning: Do different neural networks learn the same representations?cited by 220
  7. 2021Revisiting Model Stitching to Compare Neural Representationscited by 209
  8. 2014Object Detectors Emerge in Deep Scene CNNscited by 179
  9. 2018Learning Latent Permutations with Gumbel-Sinkhorn Networkscited by 175
  10. 2021Generalized Shape Metrics on Neural Representationscited by 112
  11. 2016Topology and Geometry of Half-Rectified Network Optimizationcited by 102
  12. 2020Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamicscited by 96
  13. 2020Optimizing Mode Connectivity via Neuron Alignmentcited by 86
  14. 2019Reverse-Engineering Deep ReLU Networkscited by 83
  15. 2021Similarity and Matching of Neural Network Representationscited by 78
  16. 2019On the Expressive Power of Deep Polynomial Neural Networkscited by 66
  17. 2020SoftSort: A Continuous Relaxation for the argsort Operatorcited by 66
  18. 2019Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscapecited by 63
  19. 2018Towards Understanding Learning Representations: To What Extent Do Different Neural Networks Learn the Same Representationcited by 62
  20. 2021Grounding Representation Similarity with Statistical Testingcited by 29
  21. 2018G-SGD: Optimizing ReLU Neural Networks in its Positively Scale-Invariant Spacecited by 27
  22. 2019Discovery of Natural Language Concepts in Individual Units of CNNscited by 21
  23. 2015Understanding symmetries in deep networkscited by 17
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  25. 2019Positively Scale-Invariant Flatness of ReLU Neural Networkscited by 11
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 12; 2024: 12; 2025: 18; 2026: 19.

64 papers.

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  1. 2022Git Re-Basin: Merging Models modulo Permutation Symmetriescited by 471
  2. 2023Similarity of Neural Network Models: A Survey of Functional and Representational Measurescited by 131
  3. 2022Toward Transparent AI: A Survey on Interpreting the Inner Structures of Deep Neural Networkscited by 129
  4. 2023Equivariant Architectures for Learning in Deep Weight Spacescited by 107
  5. 2024Universal Neurons in GPT2 Language Modelscited by 102
  6. 2023Deep Model Fusion: A Surveycited by 91
  7. 2023Permutation Equivariant Neural Functionalscited by 70
  8. 2023Graph Metanetworks for Processing Diverse Neural Architecturescited by 57
  9. 2023Transformer Fusion with Optimal Transportcited by 36
  10. 2025Symmetry in Neural Network Parameter Spacescited by 36
  11. 2024Universal Neural Functionalscited by 26
  12. 2024Scale Equivariant Graph Metanetworkscited by 24
38 more
  1. 2024Locking Down the Finetuned LLMs Safetycited by 22
  2. 2024The Empirical Impact of Neural Parameter Symmetries, or Lack Thereofcited by 21
  3. 2024Monomial Matrix Group Equivariant Neural Functional Networkscited by 17
  4. 2025Beyond the Permutation Symmetry of Transformers: The Role of Rotation for Model Fusioncited by 16
  5. 2025Domain-Specific Pruning of Large Mixture-of-Experts Models with Few-shot Demonstrationscited by 16
  6. 2023Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theorycited by 12
  7. 2023A Symmetry-Aware Exploration of Bayesian Neural Network Posteriorscited by 12
  8. 2023Understanding the Inner Workings of Language Models Through Representation Dissimilaritycited by 12
  9. 2025Parameter Symmetry Potentially Unifies Deep Learning Theorycited by 12
  10. 2025Identifiability of Deep Polynomial Neural Networkscited by 12
  11. 2024Neural Lineagecited by 9
  12. 2023Investigating how ReLU-networks encode symmetriescited by 8
  13. 2024Remove Symmetries to Control Model Expressivity and Improve Optimizationcited by 8
  14. 2024Unification of Symmetries Inside Neural Networks: Transformer, Feedforward and Neural ODEcited by 7
  15. 2026Identifiable Equivariant Networks are Layerwise Equivariantcited by 7
  16. 2025Model Unmerging: Making Your Models Unmergeable for Secure Model Sharingcited by 6
  17. 2025The Alexander-Hirschowitz theorem for neurovarietiescited by 6
  18. 2025Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematicscited by 5
  19. 2023On permutation symmetries in Bayesian neural network posteriors: a variational perspectivecited by 4
  20. 2024Relative Representations: Topological and Geometric Perspectivescited by 4
  21. 2025ARC: Anchored Representation Clouds for High-Resolution INR Classificationcited by 3
  22. 2026A Complete Symmetry Classification of Shallow ReLU Networkscited by 3
  23. 2022Do Neural Networks Trained with Topological Features Learn Different Internal Representations?cited by 2
  24. 2023Stress and Adaptation: Applying Anna Karenina Principle in Deep Learning for Image Classificationcited by 2
  25. 2024Weight Scope Alignment: A Frustratingly Easy Method for Model Mergingcited by 2
  26. 2025Meta Pruning via Graph Metanetworks : A Universal Meta Learning Framework for Network Pruningcited by 2
  27. 2025Emergence of Quantised Representations Isolated to Anisotropic Functionscited by 2
  28. 2025Cross-Model Semantics in Representation Learningcited by 2
  29. 2025Nonlinear dynamics of localization in neural receptive fieldscited by 1
  30. 2025The Empirical Impact of Reducing Symmetries on the Performance of Deep Ensembles and MoEcited by 1
  31. 2026Stimulus symmetries can confound representational similarity analysescited by 1
  32. 2026PAC--Bayes Bounds on Quotient Parameter Spaces: Geometry-induced Implicit-Bias Priorscited by 1
  33. 2026Not all solutions are created equal: An analytical dissociation of functional and representational similarity in deep linear neural networkscited by 1
  34. 2024Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networkscited by 0
  35. 2025Exchangeability in Neural Network and its Application to Dynamic Pruningcited by 0
  36. 2025Symmetry-Aware Graph Metanetwork Autoencoders: Model Merging through Parameter Canonicalizationcited by 0
  37. 2025Drawback of Enforcing Equivariance and its Compensation via the Lens of Expressive Powercited by 0
  38. 2025Metanetworks as Regulatory Operators: Learning to Edit for Requirement Compliancecited by 0

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