# Combining Statistical Depth and Fermat Distance for Uncertainty Quantification

arXiv: https://arxiv.org/abs/2404.08476 · HTML: https://mlsanity.com/paper/2404.08476/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 33 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Builds on (most cited first, 16 of 16)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [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 Clustering for Unsupervised Learning of Visual Features](https://mlsanity.com/paper/1807.05520/) (2018)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Out-of-Distribution Detection with Deep Nearest Neighbors](https://mlsanity.com/paper/2204.06507/) (2022)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering](https://mlsanity.com/paper/1610.04794/) (2016)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Nonparametric Uncertainty Quantification for Single Deterministic Neural Network](https://mlsanity.com/paper/2202.03101/) (2022)
- [Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms](https://mlsanity.com/paper/2307.05750/) (2023)

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

- [Data Depth as a Risk](https://mlsanity.com/paper/2507.08518/) (2025)

## Abstract (arXiv)

We measure the Out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called ``Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the ``depth'' of a point with respect to a distribution in feature space, without any assumption about the form of distribution. Our method has no trainable parameter. The method is applicable to any classification model as it is applied directly in feature space at test time and does not intervene in training process. As such, it does not impact the performance of the original model. The proposed method gives excellent qualitative result on toy datasets and can give competitive or better uncertainty estimation on standard deep learning datasets compared to strong baseline methods.