# Topological Measurement of Deep Neural Networks Using Persistent Homology

arXiv: https://arxiv.org/abs/2106.03016 · HTML: https://mlsanity.com/paper/2106.03016/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [Evaluating the visualization of what a Deep Neural Network has learned](https://mlsanity.com/paper/1509.06321/) (2015)
- [Unmasking Clever Hans Predictors and Assessing What Machines Really Learn](https://mlsanity.com/paper/1902.10178/) (2019)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/1802.04443/) (2018)
- [Convolutional Neural Fabrics](https://mlsanity.com/paper/1606.02492/) (2016)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [The Simplex Tree: an Efficient Data Structure for General Simplicial Complexes](https://mlsanity.com/paper/2001.02581/) (2020)
- [Cliques and Cavities in the Human Connectome](https://mlsanity.com/paper/1608.03520/) (2016)
- [What can topology tell us about the neural code?](https://mlsanity.com/paper/1605.01905/) (2016)
- [The topology of the directed clique complex as a network invariant](https://mlsanity.com/paper/1510.00660/) (2015)

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

- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Topological Perspectives on Optimal Multimodal Embedding Spaces](https://mlsanity.com/paper/2405.18867/) (2024)
- [Functional Network: A Novel Framework for Interpretability of Deep Neural Networks](https://mlsanity.com/paper/2205.11702/) (2022)
- [Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds](https://mlsanity.com/paper/2402.15058/) (2024)
- [Topological Signatures of Grokking](https://mlsanity.com/paper/2605.06352/) (2026)
- [HOLE: Homological Observation of Latent Embeddings for Neural Network Interpretability](https://mlsanity.com/paper/2512.07988/) (2025)
- [Empirical Evidence for Simply Connected Decision Regions in Image Classifiers](https://mlsanity.com/paper/2605.06380/) (2026)
- [Topological Simplification in Predictive Coding Networks](https://mlsanity.com/paper/2608.02816/) (2026)

## Abstract (arXiv)

The inner representation of deep neural networks (DNNs) is indecipherable, which makes it difficult to tune DNN models, control their training process, and interpret their outputs. In this paper, we propose a novel approach to investigate the inner representation of DNNs through topological data analysis (TDA). Persistent homology (PH), one of the outstanding methods in TDA, was employed for investigating the complexities of trained DNNs. We constructed clique complexes on trained DNNs and calculated the one-dimensional PH of DNNs. The PH reveals the combinational effects of multiple neurons in DNNs at different resolutions, which is difficult to be captured without using PH. Evaluations were conducted using fully connected networks (FCNs) and networks combining FCNs and convolutional neural networks (CNNs) trained on the MNIST and CIFAR-10 data sets. Evaluation results demonstrate that the PH of DNNs reflects both the excess of neurons and problem difficulty, making PH one of the prominent methods for investigating the inner representation of DNNs.