# Topological Data Analysis of Decision Boundaries with Application to Model Selection

arXiv: https://arxiv.org/abs/1805.09949 · HTML: https://mlsanity.com/paper/1805.09949/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2018.

## Built on by (most cited first, 19 of 19)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [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)
- [HardVis: Visual Analytics to Handle Instance Hardness Using Undersampling and Oversampling Techniques](https://mlsanity.com/paper/2203.15753/) (2022)
- [Deep neural networks architectures from the perspective of manifold learning](https://mlsanity.com/paper/2306.03406/) (2023)
- [Unveiling the Hessian's Connection to the Decision Boundary](https://mlsanity.com/paper/2306.07104/) (2023)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [Zeroth-Order Topological Insights into Iterative Magnitude Pruning](https://mlsanity.com/paper/2206.06563/) (2022)
- [Do Neural Networks Trained with Topological Features Learn Different Internal Representations?](https://mlsanity.com/paper/2211.07697/) (2022)
- [A Theoretical Study of Neural Network Expressive Power via Manifold Topology](https://mlsanity.com/paper/2410.16542/) (2024)
- [Addressing caveats of neural persistence with deep graph persistence](https://mlsanity.com/paper/2307.10865/) (2023)
- [Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require](https://mlsanity.com/paper/2609.08561/) (2026)
- [Concept Boundary Vectors](https://mlsanity.com/paper/2412.15698/) (2024)
- [Understanding the geometry of deep learning with decision boundary volume](https://mlsanity.com/paper/2603.14768/) (2026)
- [Empirical Evidence for Simply Connected Decision Regions in Image Classifiers](https://mlsanity.com/paper/2605.06380/) (2026)
- [Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations](https://mlsanity.com/paper/2606.07627/) (2026)

## Abstract (arXiv)

We propose the labeled vCech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples. Our main objective is quantification of deep neural network complexity to enable matching of datasets to pre-trained models; we report results for experiments using MNIST, FashionMNIST, and CIFAR10.