# Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey

arXiv: https://arxiv.org/abs/2312.05840 · HTML: https://mlsanity.com/paper/2312.05840/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 148 references that are in the map.
- First posted on arXiv 10 Dec 2023.

## Builds on (most cited first, 14 of 14)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [NAS-Bench-NLP: Neural Architecture Search Benchmark for Natural Language Processing](https://mlsanity.com/paper/2006.07116/) (2020)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [NeurIPS 2020 Competition: Predicting Generalization in Deep Learning](https://mlsanity.com/paper/2012.07976/) (2020)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [An Introduction to Multiparameter Persistence](https://mlsanity.com/paper/2203.14289/) (2022)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Disentanglement Learning via Topology](https://mlsanity.com/paper/2308.12696/) (2023)

## Built on by (most cited first, 9 of 9)

- [TOAST: Transformer Optimization using Adaptive and Simple Transformations](https://mlsanity.com/paper/2410.04941/) (2024)
- [Topological Signatures of Grokking](https://mlsanity.com/paper/2605.06352/) (2026)
- [Topology of Out-of-Distribution Examples in Deep Neural Networks](https://mlsanity.com/paper/2501.12522/) (2025)
- [Topology-Aware Layer Pruning for Large Vision-Language Models](https://mlsanity.com/paper/2604.16502/) (2026)
- [Topology-Enhanced Alignment for Large Language Models: Trajectory Topology Loss and Topological Preference Optimization](https://mlsanity.com/paper/2605.07172/) (2026)
- [How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule](https://mlsanity.com/paper/2608.01268/) (2026)
- [Measuring Memory and Generalization as Separable Geometric Channels: The Topo^2 Framework](https://mlsanity.com/paper/2608.30487/) (2026)
- [Skeletal Prototypes on Iterative Nerve Expansions](https://mlsanity.com/paper/2609.16170/) (2026)
- [Looking Inside LLMs: Small-World Connectivity as a Signature of Reasoning Performance](https://mlsanity.com/paper/2610.12304/) (2026)

## Abstract (arXiv)

This survey provides a comprehensive exploration of applications of Topological Data Analysis (TDA) within neural network analysis. Using TDA tools such as persistent homology and Mapper, we delve into the intricate structures and behaviors of neural networks and their datasets. We discuss different strategies to obtain topological information from data and neural networks by means of TDA. Additionally, we review how topological information can be leveraged to analyze properties of neural networks, such as their generalization capacity or expressivity. We explore practical implications of deep learning, specifically focusing on areas like adversarial detection and model selection. Our survey organizes the examined works into four broad domains: 1. Characterization of neural network architectures; 2. Analysis of decision regions and boundaries; 3. Study of internal representations, activations, and parameters; 4. Exploration of training dynamics and loss functions. Within each category, we discuss several articles, offering background information to aid in understanding the various methodologies. We conclude with a synthesis of key insights gained from our study, accompanied by a discussion of challenges and potential advancements in the field.