mlsanityData to 13 Oct 2026

A mathematical theory for understanding when abstract representations emerge in neural networks

arXiv 2510.09816 ↗First posted 10 Oct 2025q-bio.NC math.OC physics.bio-phcited by 4

Not on the map. Why?

Abstract (arXiv)

Recent experiments in neuroscience reveal that task-relevant variables are often encoded in approximately orthogonal subspaces of neural population activity. These disentangled, or abstract, representations have been observed in multiple brain areas and across different species. These representations have been shown to support out of distribution generalization and rapid learning of novel tasks. The mechanisms by which these representations emerge remain poorly understood, especially in the case of supervised task behavior. Here, we show mathematically that abstract representations of latent variables are guaranteed to appear in the hidden layer of feedforward nonlinear networks when they are trained on tasks that depend directly on these latent variables. These learned abstract representations reflect the semantics of the input stimuli. To show this, we reformulate the usual optimization over the network weights into a mean field optimization problem over the distribution of neural preactivations. We then apply this framework to finite-width ReLU networks and show that the hidden layer of these networks will exhibit an abstract representation at all global minima of the task objective. Finally, we extend our findings to two broad families of activation functions as well as deep feedforward architectures. Together, our results provide an explanation for the widely observed abstract representations in both the brain and artificial neural networks. In addition, the general framework that we develop here provides a mathematically tractable toolkit for understanding the emergence of different kinds of representations in task-optimized, feature-learning network models.

Key facts. Cited by 4 papers in the map (data through 13 Oct 2026). Builds on 37 of its 106 references that are in the map. First posted on arXiv 10 Oct 2025.

Builds on

37 of 106 references.

References · 37

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,610
  2. 2021Convex optimizationcited by 2,488
  3. 2019Similarity of Neural Network Representations Revisitedcited by 1,719
  4. 2018Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representationscited by 934
  5. 2017An Overview of Multi-Task Learning in Deep Neural Networkscited by 799
  6. 2013Exact solutions to the nonlinear dynamics of learning in deep linear neural networkscited by 755
  7. 2020Prevalence of Neural Collapse during the terminal phase of deep learning trainingcited by 716
  8. 2017Towards Principled Methods for Training Generative Adversarial Networkscited by 464
  9. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 425
  10. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
  11. 2020Kernel and Rich Regimes in Overparametrized Modelscited by 305
  12. 2021A Geometric Analysis of Neural Collapse with Unconstrained Featurescited by 226
25 more
  1. 2015Convergent Learning: Do different neural networks learn the same representations?cited by 215
  2. 2018A mathematical theory of semantic development in deep neural networkscited by 184
  3. 2020Tensor Programs II: Neural Tangent Kernel for Any Architecturecited by 141
  4. 2018Recent Advances in Autoencoder-Based Representation Learningcited by 131
  5. 2021Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariancescited by 127
  6. 2022Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networkscited by 104
  7. 2022Extended Unconstrained Features Model for Exploring Deep Neural Collapsecited by 102
  8. 2022Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputscited by 75
  9. 2020Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Kernel Renormalizationcited by 71
  10. 2018Gradient Descent Quantizes ReLU Network Featurescited by 59
  11. 2020A Sober Look at the Unsupervised Learning of Disentangled Representations and their Evaluationcited by 52
  12. 2019On Connected Sublevel Sets in Deep Learningcited by 51
  13. 2022The neuroconnectionist research programmecited by 48
  14. 2023Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Modelcited by 43
  15. 2022Disentanglement with Biological Constraints: A Theory of Functional Cell Typescited by 30
  16. 2020Artificial neural networks for neuroscientists: A primercited by 26
  17. 2024Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapsecited by 16
  18. 2025Adaptive kernel predictors from feature-learning infinite limits of neural networkscited by 13
  19. 2022Disentangling representations in Restricted Boltzmann Machines without adversariescited by 11
  20. 2024Coding schemes in neural networks learning classification taskscited by 11
  21. 2024Neural population geometry and optimal coding of tasks with shared latent structurecited by 9
  22. 2024Task structure and nonlinearity jointly determine learned representational geometrycited by 6
  23. 2023The RL Perceptron: Generalisation Dynamics of Policy Learning in High Dimensionscited by 5
  24. 2024Range, not Independence, Drives Modularity in Biologically Inspired Representationscited by 4
  25. 2025Compositional Generalization via Forced Rendering of Disentangled Latentscited by 4

Built on by

Cited by year (2026 to 13 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026Task Ecologies and the Evolution of World-Tracking Representations in Large Language Modelscited by 0
  2. 2026Optimal Representation Size: High-Dimensional Analysis of Pretraining and Linear Probingcited by 0
  3. 2026How are linear representations learned? Exact solutions to the dynamics of abstractioncited by 0
  4. 2026Small transformers track Bayesian evidence for latent common causes via a context-invariant mechanismcited by 0