# Approximations to the Fisher Information Metric of Deep Generative Models for Out-Of-Distribution Detection

arXiv: https://arxiv.org/abs/2403.01485 · HTML: https://mlsanity.com/paper/2403.01485/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 37 of its 55 references that are in the map.
- First posted on arXiv 3 Mar 2024.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 30 of 37)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Deep Anomaly Detection with Outlier Exposure](https://mlsanity.com/paper/1812.04606/) (2018)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [Input complexity and out-of-distribution detection with likelihood-based generative models](https://mlsanity.com/paper/1909.11480/) (2019)
- [Autoencoders for Unsupervised Anomaly Segmentation in Brain MR Images: A Comparative Study](https://mlsanity.com/paper/2004.03271/) (2020)
- [Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis](https://mlsanity.com/paper/1806.03884/) (2018)
- [Likelihood Regret: An Out-of-Distribution Detection Score For Variational Auto-encoder](https://mlsanity.com/paper/2003.02977/) (2020)
- [Denoising diffusion models for out-of-distribution detection](https://mlsanity.com/paper/2211.07740/) (2022)
- [Understanding Failures in Out-of-Distribution Detection with Deep Generative Models](https://mlsanity.com/paper/2107.06908/) (2021)
- [Density of States Estimation for Out-of-Distribution Detection](https://mlsanity.com/paper/2006.09273/) (2020)
- [Understanding Anomaly Detection with Deep Invertible Networks through Hierarchies of Distributions and Features](https://mlsanity.com/paper/2006.10848/) (2020)
- [Hierarchical VAEs Know What They Don't Know](https://mlsanity.com/paper/2102.08248/) (2021)
- [Backpropagated Gradient Representations for Anomaly Detection](https://mlsanity.com/paper/2007.09507/) (2020)
- [Trust Issues: Uncertainty Estimation Does Not Enable Reliable OOD Detection On Medical Tabular Data](https://mlsanity.com/paper/2011.03274/) (2020)
- [Perfect density models cannot guarantee anomaly detection](https://mlsanity.com/paper/2012.03808/) (2020)

## Built on by (most cited first, 2 of 2)

- [An Isotropic Approach to Efficient Uncertainty Quantification with Gradient Norms](https://mlsanity.com/paper/2603.29466/) (2026)
- [Beyond Binary Out-of-Distribution Detection: Characterizing Distributional Shifts with Multi-Statistic Diffusion Trajectories](https://mlsanity.com/paper/2510.17381/) (2025)

## Abstract (arXiv)

Likelihood-based deep generative models such as score-based diffusion models and variational autoencoders are state-of-the-art machine learning models approximating high-dimensional distributions of data such as images, text, or audio. One of many downstream tasks they can be naturally applied to is out-of-distribution (OOD) detection. However, seminal work by Nalisnick et al. which we reproduce showed that deep generative models consistently infer higher log-likelihoods for OOD data than data they were trained on, marking an open problem. In this work, we analyse using the gradient of a data point with respect to the parameters of the deep generative model for OOD detection, based on the simple intuition that OOD data should have larger gradient norms than training data. We formalise measuring the size of the gradient as approximating the Fisher information metric. We show that the Fisher information matrix (FIM) has large absolute diagonal values, motivating the use of chi-square distributed, layer-wise gradient norms as features. We combine these features to make a simple, model-agnostic and hyperparameter-free method for OOD detection which estimates the joint density of the layer-wise gradient norms for a given data point. We find that these layer-wise gradient norms are weakly correlated, rendering their combined usage informative, and prove that the layer-wise gradient norms satisfy the principle of (data representation) invariance. Our empirical results indicate that this method outperforms the Typicality test for most deep generative models and image dataset pairings.