# Scalable anomaly detection via a univariate Christoffel function

arXiv: https://arxiv.org/abs/2606.12483 · HTML: https://mlsanity.com/paper/2606.12483/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [ADBench: Anomaly Detection Benchmark](https://mlsanity.com/paper/2206.09426/) (2022)
- [ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions](https://mlsanity.com/paper/2201.00382/) (2022)
- [Scalable, Explainable and Provably Robust Anomaly Detection with One-Step Flow Matching](https://mlsanity.com/paper/2510.18328/) (2025)
- [Sorting out typicality with the inverse moment matrix SOS polynomial](https://mlsanity.com/paper/1606.03858/) (2016)
- [Rate of convergence for geometric inference based on the empirical Christoffel function](https://mlsanity.com/paper/1910.14458/) (2019)
- [Kernel-based Outlier Detection using the Inverse Christoffel Function](https://mlsanity.com/paper/1806.06775/) (2018)
- [Leveraging the Christoffel Function for Outlier Detection in Data Streams](https://mlsanity.com/paper/2508.16617/) (2025)
- [High-Dimensional Statistics: Reflections on Progress and Open Problems](https://mlsanity.com/paper/2605.05076/) (2026)
- [KNN and K-means in Gini Prametric Spaces](https://mlsanity.com/paper/2501.18028/) (2025)
- [A Christoffel-like function for high-dimensional support inference in graphical models](https://mlsanity.com/paper/2409.15965/) (2024)

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

- [CLOE: Christoffel Loss Autoencoder for Anomaly Detection](https://mlsanity.com/paper/2607.20530/) (2026)
- [Early Failure Prediction from Near-Anomaly Detection: A Proactive Approach](https://mlsanity.com/paper/2607.26704/) (2026)

## Abstract (arXiv)

Anomaly detection plays a critical role in identifying unusual patterns across domains such as fraud detection, network intrusion, and system fault diagnosis. Recently, Christoffel function-based methods, rooted in polynomial optimization, have emerged as promising alternatives to deep learning due to their strong mathematical foundations and computational frugality. However, their practical applicability is hindered by the need to invert a matrix whose size grows exponentially with the data dimension, rendering the method intractable even for moderate-dimensional datasets. This paper addresses the dimensionality limitations of Christoffel function-based anomaly detection while preserving its key theoretical properties, i.e., the on-off support dichotomy behavior and the accurate support shape capture. We introduce UCF, a univariate Christoffel function which is based on the squared distance between the query point and the support points. Extensive experiments on the ADBench benchmark demonstrate that UCF consistently outperforms 14 state-of-the-art baselines in terms of Average Precision. By resolving the scalability bottleneck of the Christoffel Function, this work expands the toolkit of anomaly detection methods with a robust, theoretically grounded, and universally applicable approach.