# The Training Process of Many Deep Networks Explores the Same Low-Dimensional Manifold

arXiv: https://arxiv.org/abs/2305.01604 · HTML: https://mlsanity.com/paper/2305.01604/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 85 references that are in the map.
- First posted on arXiv 2 May 2023.

## Builds on (most cited first, 30 of 39)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [A Baseline for Few-Shot Image Classification](https://mlsanity.com/paper/1909.02729/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Patches Are All You Need?](https://mlsanity.com/paper/2201.09792/) (2022)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)

## Built on by (most cited first, 21 of 21)

- [The Universal Weight Subspace Hypothesis](https://mlsanity.com/paper/2512.05117/) (2025)
- [Dynamics of Transient Structure in In-Context Linear Regression Transformers](https://mlsanity.com/paper/2501.17745/) (2025)
- [Density estimation with LLMs: a geometric investigation of in-context learning trajectories](https://mlsanity.com/paper/2410.05218/) (2024)
- [Narrow Fine-Tuning Erodes Safety Alignment in Vision-Language Agents](https://mlsanity.com/paper/2602.16931/) (2026)
- [Position: Weight Space Should Be a First-Class Generative AI Modality](https://mlsanity.com/paper/2605.18632/) (2026)
- [Learning Capacity: A Measure of the Effective Dimensionality of a Model](https://mlsanity.com/paper/2305.17332/) (2023)
- [Many Perception Tasks are Highly Redundant Functions of their Input Data](https://mlsanity.com/paper/2407.13841/) (2024)
- [Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds](https://mlsanity.com/paper/2603.28764/) (2026)
- [A Waddington landscape for prototype learning in generalized Hopfield networks](https://mlsanity.com/paper/2312.03012/) (2023)
- [Comparing the information content of probabilistic representation spaces](https://mlsanity.com/paper/2405.21042/) (2024)
- [Distributional Spectral Diagnostics for Localizing Grokking Transitions](https://mlsanity.com/paper/2605.08237/) (2026)
- [Circuit design in biology and machine learning. I. Random networks and dimensional reduction](https://mlsanity.com/paper/2408.09604/) (2024)
- [An Analytical Characterization of Sloppiness in Neural Networks: Insights from Linear Models](https://mlsanity.com/paper/2505.08915/) (2025)
- [Model Agreement via Anchoring](https://mlsanity.com/paper/2602.23360/) (2026)
- [Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE](https://mlsanity.com/paper/2607.07034/) (2026)
- [Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networks](https://mlsanity.com/paper/2409.19138/) (2024)
- [Contrastive Learning to Fine-Tune Feature Extraction Models for the Visual Cortex](https://mlsanity.com/paper/2410.06067/) (2024)
- [Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence](https://mlsanity.com/paper/2604.17067/) (2026)
- [Bayesian Fine-tuning in Projected Subspaces](https://mlsanity.com/paper/2605.07706/) (2026)
- [Scalar Representations of Neural Network Training Dynamics](https://mlsanity.com/paper/2606.30384/) (2026)
- [Modeling and Understanding Neural Weights of Humanoid Policies with Diffusion](https://mlsanity.com/paper/2610.20606/) (2026)

## Abstract (arXiv)

We develop information-geometric techniques to analyze the trajectories of the predictions of deep networks during training. By examining the underlying high-dimensional probabilistic models, we reveal that the training process explores an effectively low-dimensional manifold. Networks with a wide range of architectures, sizes, trained using different optimization methods, regularization techniques, data augmentation techniques, and weight initializations lie on the same manifold in the prediction space. We study the details of this manifold to find that networks with different architectures follow distinguishable trajectories but other factors have a minimal influence; larger networks train along a similar manifold as that of smaller networks, just faster; and networks initialized at very different parts of the prediction space converge to the solution along a similar manifold.