# Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology

arXiv: https://arxiv.org/abs/2101.07752 · HTML: https://mlsanity.com/paper/2101.07752/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 19 Jan 2021.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Computing the Testing Error without a Testing Set](https://mlsanity.com/paper/2005.00450/) (2020)
- [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/1802.04443/) (2018)
- [Persistence Homology of Networks: Methods and Applications](https://mlsanity.com/paper/1907.08708/) (2019)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [Functional Summaries of Persistence Diagrams](https://mlsanity.com/paper/1804.01618/) (2018)
- [Topological Data Analysis of Decision Boundaries with Application to Model Selection](https://mlsanity.com/paper/1805.09949/) (2018)
- [Impact of Fully Connected Layers on Performance of Convolutional Neural Networks for Image Classification](https://mlsanity.com/paper/1902.02771/) (2019)
- [Topological Machine Learning with Persistence Indicator Functions](https://mlsanity.com/paper/1907.13496/) (2019)
- [Path homologies of deep feedforward networks](https://mlsanity.com/paper/1910.07617/) (2019)
- [Computing persistent homology of directed flag complexes](https://mlsanity.com/paper/1906.10458/) (2019)
- [An empirical study of the relation between network architecture and complexity](https://mlsanity.com/paper/1911.04120/) (2019)
- [Capacity allocation analysis of neural networks: A tool for principled architecture design](https://mlsanity.com/paper/1902.04485/) (2019)
- [Some Geometrical and Topological Properties of DNNs' Decision Boundaries](https://mlsanity.com/paper/2003.03687/) (2020)

## Built on by (most cited first, 8 of 8)

- [Generalization Bounds with Data-dependent Fractal Dimensions](https://mlsanity.com/paper/2302.02766/) (2023)
- [Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set](https://mlsanity.com/paper/2106.00012/) (2021)
- [Estimating class separability of text embeddings with persistent homology](https://mlsanity.com/paper/2305.15016/) (2023)
- [Leveraging The Topological Consistencies of Learning in Deep Neural Networks](https://mlsanity.com/paper/2111.15651/) (2021)
- [Characterization of topological structures in different neural network architectures](https://mlsanity.com/paper/2407.06286/) (2024)
- [Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning](https://mlsanity.com/paper/2512.08241/) (2025)
- [From Layers to Networks: Comparing Neural Representations via Diffusion Geometry](https://mlsanity.com/paper/2605.15901/) (2026)
- [TopoTuner: Topological Finetuning of Large Language Models](https://mlsanity.com/paper/2607.16637/) (2026)

## Abstract (arXiv)

Characterizing the structural properties of neural networks is crucial yet poorly understood, and there are no well-established similarity measures between networks. In this work, we observe that neural networks can be represented as abstract simplicial complex and analyzed using their topological 'fingerprints' via Persistent Homology (PH). We then describe a PH-based representation proposed for characterizing and measuring similarity of neural networks. We empirically show the effectiveness of this representation as a descriptor of different architectures in several datasets. This approach based on Topological Data Analysis is a step towards better understanding neural networks and serves as a useful similarity measure.