# Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology

arXiv: https://arxiv.org/abs/1812.09764 · HTML: https://mlsanity.com/paper/1812.09764/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2018.

## Built on by (most cited first, 30 of 73)

- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Artificial Text Detection via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2109.04825/) (2021)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Learning Global Hypothesis Space for Enhancing Synergistic Reasoning Chain](https://mlsanity.com/paper/2602.09794/) (2026)
- [A survey on shape-constraint deep learning for medical image segmentation](https://mlsanity.com/paper/2101.07721/) (2021)
- [Generalization Bounds with Data-dependent Fractal Dimensions](https://mlsanity.com/paper/2302.02766/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [Experimental Observations of the Topology of Convolutional Neural Network Activations](https://mlsanity.com/paper/2212.00222/) (2022)
- [ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog](https://mlsanity.com/paper/2306.17418/) (2023)
- [Topological Measurement of Deep Neural Networks Using Persistent Homology](https://mlsanity.com/paper/2106.03016/) (2021)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Selecting which Dense Retriever to use for Zero-Shot Search](https://mlsanity.com/paper/2309.09403/) (2023)
- [Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation](https://mlsanity.com/paper/2103.09716/) (2021)
- [BERTops: Studying BERT Representations under a Topological Lens](https://mlsanity.com/paper/2205.00953/) (2022)
- [FONDUE: an algorithm to find the optimal dimensionality of the latent representations of variational autoencoders](https://mlsanity.com/paper/2209.12806/) (2022)
- [Deep neural networks architectures from the perspective of manifold learning](https://mlsanity.com/paper/2306.03406/) (2023)
- [On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/2311.04592/) (2023)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis](https://mlsanity.com/paper/2205.05662/) (2022)
- [Representation Geometry as a Diagnostic for Out-of-Distribution Robustness](https://mlsanity.com/paper/2602.03951/) (2026)
- [Internal Representations of Vision Models Through the Lens of Frames on Data Manifolds](https://mlsanity.com/paper/2211.10558/) (2022)
- [Singular value decomposition based matrix surgery](https://mlsanity.com/paper/2302.11446/) (2023)

## Abstract (arXiv)

While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In this work, we propose neural persistence, a complexity measure for neural network architectures based on topological data analysis on weighted stratified graphs. To demonstrate the usefulness of our approach, we show that neural persistence reflects best practices developed in the deep learning community such as dropout and batch normalization. Moreover, we derive a neural persistence-based stopping criterion that shortens the training process while achieving comparable accuracies as early stopping based on validation loss.