# Topology of Out-of-Distribution Examples in Deep Neural Networks

arXiv: https://arxiv.org/abs/2501.12522 · HTML: https://mlsanity.com/paper/2501.12522/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 21 Jan 2025.

## Builds on (most cited first, 21 of 21)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Deep Learning for Medical Image Analysis](https://mlsanity.com/paper/1708.08987/) (2017)
- [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)
- [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)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [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)
- [Confidence sets for persistence diagrams](https://mlsanity.com/paper/1303.7117/) (2013)
- [Surround-View Vision-based 3D Detection for Autonomous Driving: A Survey](https://mlsanity.com/paper/2302.06650/) (2023)
- [Adversary Detection in Neural Networks via Persistent Homology](https://mlsanity.com/paper/1711.10056/) (2017)
- [Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey](https://mlsanity.com/paper/2312.05840/) (2023)
- [On the Bootstrap for Persistence Diagrams and Landscapes](https://mlsanity.com/paper/1311.0376/) (2013)
- [Importance attribution in neural networks by means of persistence landscapes of time series](https://mlsanity.com/paper/2302.03132/) (2023)

## Built on by (most cited first, 1 of 1)

- [MAD-OOD: A Deep Learning Cluster-Driven Framework for an Out-of-Distribution Malware Detection and Classification](https://mlsanity.com/paper/2512.17594/) (2025)

## Abstract (arXiv)

As deep neural networks (DNNs) become increasingly common, concerns about their robustness do as well. A longstanding problem for deployed DNNs is their behavior in the face of unfamiliar inputs; specifically, these models tend to be overconfident and incorrect when encountering out-of-distribution (OOD) examples. In this work, we present a topological approach to characterizing OOD examples using latent layer embeddings from DNNs. Our goal is to identify topological features, referred to as landmarks, that indicate OOD examples. We conduct extensive experiments on benchmark datasets and a realistic DNN model, revealing a key insight for OOD detection. Well-trained DNNs have been shown to induce a topological simplification on training data for simple models and datasets; we show that this property holds for realistic, large-scale test and training data, but does not hold for OOD examples. More specifically, we find that the average lifetime (or persistence) of OOD examples is statistically longer than that of training or test examples. This indicates that DNNs struggle to induce topological simplification on unfamiliar inputs. Our empirical results provide novel evidence of topological simplification in realistic DNNs and lay the groundwork for topologically-informed OOD detection strategies.