# Outlier Detection through Null Space Analysis of Neural Networks

arXiv: https://arxiv.org/abs/2007.01263 · HTML: https://mlsanity.com/paper/2007.01263/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2020.

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

- [ViM: Out-Of-Distribution with Virtual-logit Matching](https://mlsanity.com/paper/2203.10807/) (2022)
- [NECO: NEural Collapse Based Out-of-distribution detection](https://mlsanity.com/paper/2310.06823/) (2023)
- [Simplicity Bias in 1-Hidden Layer Neural Networks](https://mlsanity.com/paper/2302.00457/) (2023)
- [Pursuing Feature Separation based on Neural Collapse for Out-of-Distribution Detection](https://mlsanity.com/paper/2405.17816/) (2024)
- [WDiscOOD: Out-of-Distribution Detection via Whitened Linear Discriminant Analysis](https://mlsanity.com/paper/2303.07543/) (2023)
- [SFP: Spurious Feature-targeted Pruning for Out-of-Distribution Generalization](https://mlsanity.com/paper/2305.11615/) (2023)
- [Low-Dimensional Gradient Helps Out-of-Distribution Detection](https://mlsanity.com/paper/2310.17163/) (2023)
- [Sonar-based Deep Learning in Underwater Robotics: Overview, Robustness and Challenges](https://mlsanity.com/paper/2412.11840/) (2024)
- [Quantifying Overfitting: Evaluating Neural Network Performance through Analysis of Null Space](https://mlsanity.com/paper/2305.19424/) (2023)
- [NERO: Explainable Out-of-Distribution Detection with Neuron-level Relevance](https://mlsanity.com/paper/2506.15404/) (2025)
- [Null Space Properties of Neural Networks with Applications to Image Steganography](https://mlsanity.com/paper/2401.10262/) (2024)
- [FEVER-OOD: Free Energy Vulnerability Elimination for Robust Out-of-Distribution Detection](https://mlsanity.com/paper/2412.01596/) (2024)
- [Activation Subspaces for Out-of-Distribution Detection](https://mlsanity.com/paper/2508.21695/) (2025)
- [Improving Out-of-Distribution Detection with Disentangled Foreground and Background Features](https://mlsanity.com/paper/2303.08727/) (2023)
- [Dimensionality-induced information loss of outliers in deep neural networks](https://mlsanity.com/paper/2410.21656/) (2024)
- [Pseudo-label Induced Subspace Representation Learning for Robust Out-of-Distribution Detection](https://mlsanity.com/paper/2508.03108/) (2025)
- [The Invisible Gorilla Effect in Out-of-distribution Detection](https://mlsanity.com/paper/2602.20068/) (2026)
- [Make it SING: Analyzing Semantic Invariants in Classifiers](https://mlsanity.com/paper/2603.14610/) (2026)
- [What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry](https://mlsanity.com/paper/2607.13046/) (2026)

## Abstract (arXiv)

Many machine learning classification systems lack competency awareness. Specifically, many systems lack the ability to identify when outliers (e.g., samples that are distinct from and not represented in the training data distribution) are being presented to the system. The ability to detect outliers is of practical significance since it can help the system behave in an reasonable way when encountering unexpected data. In prior work, outlier detection is commonly carried out in a processing pipeline that is distinct from the classification model. Thus, for a complete system that incorporates outlier detection and classification, two models must be trained, increasing the overall complexity of the approach. In this paper we use the concept of the null space to integrate an outlier detection method directly into a neural network used for classification. Our method, called Null Space Analysis (NuSA) of neural networks, works by computing and controlling the magnitude of the null space projection as data is passed through a network. Using these projections, we can then calculate a score that can differentiate between normal and abnormal data. Results are shown that indicate networks trained with NuSA retain their classification performance while also being able to detect outliers at rates similar to commonly used outlier detection algorithms.