# Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set

arXiv: https://arxiv.org/abs/2106.00012 · HTML: https://mlsanity.com/paper/2106.00012/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 31 May 2021.

## Builds on (most cited first, 10 of 10)

- [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)
- [Predicting the Generalization Gap in Deep Networks with Margin Distributions](https://mlsanity.com/paper/1810.00113/) (2018)
- [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)
- [Functional Summaries of Persistence Diagrams](https://mlsanity.com/paper/1804.01618/) (2018)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Path homologies of deep feedforward networks](https://mlsanity.com/paper/1910.07617/) (2019)
- [Leave Zero Out: Towards a No-Cross-Validation Approach for Model Selection](https://mlsanity.com/paper/2012.13309/) (2020)
- [An empirical study of the relation between network architecture and complexity](https://mlsanity.com/paper/1911.04120/) (2019)

## Built on by (most cited first, 4 of 4)

- [Estimating class separability of text embeddings with persistent homology](https://mlsanity.com/paper/2305.15016/) (2023)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [HOLE: Homological Observation of Latent Embeddings for Neural Network Interpretability](https://mlsanity.com/paper/2512.07988/) (2025)
- [TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selection](https://mlsanity.com/paper/2605.08870/) (2026)

## Abstract (arXiv)

The training of neural networks is usually monitored with a validation (holdout) set to estimate the generalization of the model. This is done instead of measuring intrinsic properties of the model to determine whether it is learning appropriately. In this work, we suggest studying the training of neural networks with Algebraic Topology, specifically Persistent Homology (PH). Using simplicial complex representations of neural networks, we study the PH diagram distance evolution on the neural network learning process with different architectures and several datasets. Results show that the PH diagram distance between consecutive neural network states correlates with the validation accuracy, implying that the generalization error of a neural network could be intrinsically estimated without any holdout set.