# Relative stability toward diffeomorphisms indicates performance in deep nets

arXiv: https://arxiv.org/abs/2105.02468 · HTML: https://mlsanity.com/paper/2105.02468/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 78 references that are in the map.
- First posted on arXiv 6 May 2021.

## Builds on (most cited first, 30 of 47)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [An End-to-End Trainable Neural Network for Image-based Sequence Recognition and Its Application to Scene Text Recognition](https://mlsanity.com/paper/1507.05717/) (2015)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Deep Speech 2: End-to-End Speech Recognition in English and Mandarin](https://mlsanity.com/paper/1512.02595/) (2015)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [A Fourier Perspective on Model Robustness in Computer Vision](https://mlsanity.com/paper/1906.08988/) (2019)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [Spatially Transformed Adversarial Examples](https://mlsanity.com/paper/1801.02612/) (2018)
- [Sensitivity and Generalization in Neural Networks: an Empirical Study](https://mlsanity.com/paper/1802.08760/) (2018)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [Strike (with) a Pose: Neural Networks Are Easily Fooled by Strange Poses of Familiar Objects](https://mlsanity.com/paper/1811.11553/) (2018)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)

## Built on by (most cited first, 12 of 12)

- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [PRIME: A few primitives can boost robustness to common corruptions](https://mlsanity.com/paper/2112.13547/) (2021)
- [Learning sparse features can lead to overfitting in neural networks](https://mlsanity.com/paper/2206.12314/) (2022)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [What Can Be Learnt With Wide Convolutional Neural Networks?](https://mlsanity.com/paper/2208.01003/) (2022)
- [On the Sample Complexity of Learning under Invariance and Geometric Stability](https://mlsanity.com/paper/2106.07148/) (2021)
- [Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature Learning](https://mlsanity.com/paper/2402.08010/) (2024)
- [How Deep Networks Learn Sparse and Hierarchical Data: the Sparse Random Hierarchy Model](https://mlsanity.com/paper/2404.10727/) (2024)
- [Synergy and Symmetry in Deep Learning: Interactions between the Data, Model, and Inference Algorithm](https://mlsanity.com/paper/2207.04612/) (2022)
- [How deep convolutional neural networks lose spatial information with training](https://mlsanity.com/paper/2210.01506/) (2022)
- [Data augmentation with mixtures of max-entropy transformations for filling-level classification](https://mlsanity.com/paper/2203.04027/) (2022)
- [Measuring dissimilarity with diffeomorphism invariance](https://mlsanity.com/paper/2202.05614/) (2022)

## Abstract (arXiv)

Understanding why deep nets can classify data in large dimensions remains a challenge. It has been proposed that they do so by becoming stable to diffeomorphisms, yet existing empirical measurements support that it is often not the case. We revisit this question by defining a maximum-entropy distribution on diffeomorphisms, that allows to study typical diffeomorphisms of a given norm. We confirm that stability toward diffeomorphisms does not strongly correlate to performance on benchmark data sets of images. By contrast, we find that the stability toward diffeomorphisms relative to that of generic transformations Rf correlates remarkably with the test error εt. It is of order unity at initialization but decreases by several decades during training for state-of-the-art architectures. For CIFAR10 and 15 known architectures, we find εt≈ 0.2√Rf, suggesting that obtaining a small Rf is important to achieve good performance. We study how Rf depends on the size of the training set and compare it to a simple model of invariant learning.