# Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation

arXiv: https://arxiv.org/abs/2303.05161 · HTML: https://mlsanity.com/paper/2303.05161/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 67 references that are in the map.
- First posted on arXiv 9 Mar 2023.

## Builds on (most cited first, 30 of 31)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [Deep metric learning using Triplet network](https://mlsanity.com/paper/1412.6622/) (2014)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Analyzing and Improving Representations with the Soft Nearest Neighbor Loss](https://mlsanity.com/paper/1902.01889/) (2019)
- [Deep Learning of Representations: Looking Forward](https://mlsanity.com/paper/1305.0445/) (2013)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Where is the Information in a Deep Neural Network?](https://mlsanity.com/paper/1905.12213/) (2019)
- [Dimensionality compression and expansion in Deep Neural Networks](https://mlsanity.com/paper/1906.00443/) (2019)
- [Data-driven emergence of convolutional structure in neural networks](https://mlsanity.com/paper/2202.00565/) (2022)
- [How deep is deep enough? -- Quantifying class separability in the hidden layers of deep neural networks](https://mlsanity.com/paper/1811.01753/) (2018)
- [Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior](https://mlsanity.com/paper/1710.09553/) (2017)
- [Learning through atypical "phase transitions" in overparameterized neural networks](https://mlsanity.com/paper/2110.00683/) (2021)
- [Semi-Supervised Learning via Compact Latent Space Clustering](https://mlsanity.com/paper/1806.02679/) (2018)
- [The activity-weight duality in feed forward neural networks: The geometric determinants of generalization](https://mlsanity.com/paper/2203.10736/) (2022)
- [Intrinsic dimension estimation for locally undersampled data](https://mlsanity.com/paper/1906.07670/) (2019)
- [Linear Classification of Neural Manifolds with Correlated Variability](https://mlsanity.com/paper/2211.14961/) (2022)

## Built on by (most cited first, 6 of 6)

- [Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks](https://mlsanity.com/paper/2307.11807/) (2023)
- [Feature learning in finite-width Bayesian deep linear networks with multiple outputs and convolutional layers](https://mlsanity.com/paper/2406.03260/) (2024)
- [Reasoning emerges from constrained inference manifolds in large language models](https://mlsanity.com/paper/2605.08142/) (2026)
- [Spring-block theory of feature learning in deep neural networks](https://mlsanity.com/paper/2407.19353/) (2024)
- [The Advantage of Fine-Grained Training](https://mlsanity.com/paper/2509.05130/) (2025)
- [Kernel Renormalization in Bayesian Deep Neural Networks: the Equivalent Wishart Ansatz in the Proportional Regime](https://mlsanity.com/paper/2605.29684/) (2026)

## Abstract (arXiv)

To achieve near-zero training error in a classification problem, the layers of a feed-forward network have to disentangle the manifolds of data points with different labels, to facilitate the discrimination. However, excessive class separation can bring to overfitting since good generalisation requires learning invariant features, which involve some level of entanglement. We report on numerical experiments showing how the optimisation dynamics finds representations that balance these opposing tendencies with a non-monotonic trend. After a fast segregation phase, a slower rearrangement (conserved across data sets and architectures) increases the class entanglement.The training error at the inversion is stable under subsampling, and across network initialisations and optimisers, which characterises it as a property solely of the data structure and (very weakly) of the architecture. The inversion is the manifestation of tradeoffs elicited by well-defined and maximally stable elements of the training set, coined ``stragglers'', particularly influential for generalisation.