Scalable anomaly detection via a univariate Christoffel function

2026-06-10

AI Summary

Traditional anomaly detection methods based on the Christoffel function suffer from computational intractability in moderate to high dimensions due to the exponential growth in matrix inversion complexity. This work proposes the Univariate Christoffel Function (UCF) approach, which for the first time reduces the multivariate Christoffel function to a univariate form dependent solely on the squared distances between query points and support points. This reformulation preserves the precise geometric characterization of the support set and the distinctive “on-off” separation capability while drastically lowering computational cost. By integrating distance embedding with polynomial optimization techniques, UCF achieves state-of-the-art performance across the ADBench benchmark, outperforming 14 leading methods in average precision and demonstrating both high efficiency and broad applicability.

Abstract

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.

Problem

• anomaly detection
• Christoffel function
• scalability
• dimensionality
• matrix inversion

Innovation

• Christoffel function
• anomaly detection
• scalability
• univariate approximation
• polynomial optimization