# A Radon-Nikod'ym Perspective on Anomaly Detection: Theory and Implications

arXiv: https://arxiv.org/abs/2502.18002 · HTML: https://mlsanity.com/paper/2502.18002/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 25 Feb 2025.

## Builds on (most cited first, 18 of 18)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Revisiting Deep Learning Models for Tabular Data](https://mlsanity.com/paper/2106.11959/) (2021)
- [Real-world Anomaly Detection in Surveillance Videos](https://mlsanity.com/paper/1801.04264/) (2018)
- [Unsupervised Anomaly Detection with Generative Adversarial Networks to Guide Marker Discovery](https://mlsanity.com/paper/1703.05921/) (2017)
- [Graph Neural Network-Based Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2106.06947/) (2021)
- [Deep Semi-Supervised Anomaly Detection](https://mlsanity.com/paper/1906.02694/) (2019)
- [Progressive Image Deraining Networks: A Better and Simpler Baseline](https://mlsanity.com/paper/1901.09221/) (2019)
- [Classification-Based Anomaly Detection for General Data](https://mlsanity.com/paper/2005.02359/) (2020)
- [ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions](https://mlsanity.com/paper/2201.00382/) (2022)
- [Deep Industrial Image Anomaly Detection: A Survey](https://mlsanity.com/paper/2301.11514/) (2023)
- [COPOD: Copula-Based Outlier Detection](https://mlsanity.com/paper/2009.09463/) (2020)
- [Neural Transformation Learning for Deep Anomaly Detection Beyond Images](https://mlsanity.com/paper/2103.16440/) (2021)
- [Towards a Rigorous Evaluation of Time-series Anomaly Detection](https://mlsanity.com/paper/2109.05257/) (2021)
- [Local Evaluation of Time Series Anomaly Detection Algorithms](https://mlsanity.com/paper/2206.13167/) (2022)
- [Exathlon: A Benchmark for Explainable Anomaly Detection over Time Series](https://mlsanity.com/paper/2010.05073/) (2020)
- [An overview of deep learning based methods for unsupervised and semi-supervised anomaly detection in videos](https://mlsanity.com/paper/1801.03149/) (2018)
- [Fascinating Supervisory Signals and Where to Find Them: Deep Anomaly Detection with Scale Learning](https://mlsanity.com/paper/2305.16114/) (2023)
- [European Space Agency Benchmark for Anomaly Detection in Satellite Telemetry](https://mlsanity.com/paper/2406.17826/) (2024)

## Abstract (arXiv)

Which principle underpins the design of an effective anomaly detection loss function? The answer lies in the concept of Radon-Nikod'ym theorem, a fundamental concept in measure theory. The key insight from this article is: Multiplying the vanilla loss function with the Radon-Nikod'ym derivative improves the performance across the board. We refer to this as RN-Loss. We prove this using the setting of PAC (Probably Approximately Correct) learnability. Depending on the context a Radon-Nikod'ym derivative takes different forms. In the simplest case of supervised anomaly detection, Radon-Nikod'ym derivative takes the form of a simple weighted loss. In the case of unsupervised anomaly detection (with distributional assumptions), Radon-Nikod'ym derivative takes the form of the popular cluster based local outlier factor. We evaluate our algorithm on 96 datasets, including univariate and multivariate data from diverse domains, including healthcare, cybersecurity, and finance. We show that RN-Derivative algorithms outperform state-of-the-art methods on 68% of Multivariate datasets (based on F1 scores) and also achieves peak F1-scores on 72% of time series (Univariate) datasets.