# Topology and geometry of data manifold in deep learning

arXiv: https://arxiv.org/abs/2204.08624 · HTML: https://mlsanity.com/paper/2204.08624/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 82 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 30 of 40)

- [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)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Technical Report on the CleverHans v2.1.0 Adversarial Examples Library](https://mlsanity.com/paper/1610.00768/) (2016)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Predicting the Generalization Gap in Deep Networks with Margin Distributions](https://mlsanity.com/paper/1810.00113/) (2018)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [CelebA-Spoof: Large-Scale Face Anti-Spoofing Dataset with Rich Annotations](https://mlsanity.com/paper/2007.12342/) (2020)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [On the geometry of generalization and memorization in deep neural networks](https://mlsanity.com/paper/2105.14602/) (2021)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Deep Learning for Face Anti-Spoofing: A Survey](https://mlsanity.com/paper/2106.14948/) (2021)
- [Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks](https://mlsanity.com/paper/2006.09313/) (2020)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Computing the Testing Error without a Testing Set](https://mlsanity.com/paper/2005.00450/) (2020)
- [Generalization Error in Deep Learning](https://mlsanity.com/paper/1808.01174/) (2018)
- [The Dimpled Manifold Model of Adversarial Examples in Machine Learning](https://mlsanity.com/paper/2106.10151/) (2021)
- [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/1802.04443/) (2018)
- [Attacks on State-of-the-Art Face Recognition using Attentional Adversarial Attack Generative Network](https://mlsanity.com/paper/1811.12026/) (2018)

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

- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [Deep Learning as Ricci Flow](https://mlsanity.com/paper/2404.14265/) (2024)
- [Singular value decomposition based matrix surgery](https://mlsanity.com/paper/2302.11446/) (2023)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)
- [Neural Feature Geometry Evolves as Discrete Ricci Flow](https://mlsanity.com/paper/2509.22362/) (2025)
- [Quantifying Manifolds: Do the manifolds learned by Generative Adversarial Networks converge to the real data manifold](https://mlsanity.com/paper/2403.05033/) (2024)
- [Self-Directed Task Identification](https://mlsanity.com/paper/2604.02430/) (2026)
- [DCRL: Decoupling and Coupling Reinforcement Learning via Policy-Reward Manifold Alignment](https://mlsanity.com/paper/2609.27572/) (2026)

## Abstract (arXiv)

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of the data manifold on different layers. We also propose a method for assessing the generalizing ability of neural networks based on topological descriptors. In this paper, we use the concepts of topological data analysis and intrinsic dimension, and we present a wide range of experiments on different datasets and different configurations of convolutional neural network architectures. In addition, we consider the issue of the geometry of adversarial attacks in the classification task and spoofing attacks on face recognition systems. Our work is a contribution to the development of an important area of explainable and interpretable AI through the example of computer vision.