# CLOE: Christoffel Loss Autoencoder for Anomaly Detection

arXiv: https://arxiv.org/abs/2607.20530 · HTML: https://mlsanity.com/paper/2607.20530/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [ADBench: Anomaly Detection Benchmark](https://mlsanity.com/paper/2206.09426/) (2022)
- [PyOD: A Python Toolbox for Scalable Outlier Detection](https://mlsanity.com/paper/1901.01588/) (2019)
- [ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions](https://mlsanity.com/paper/2201.00382/) (2022)
- [Deep Learning Advancements in Anomaly Detection: A Comprehensive Survey](https://mlsanity.com/paper/2503.13195/) (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)
- [Leveraging the Christoffel Function for Outlier Detection in Data Streams](https://mlsanity.com/paper/2508.16617/) (2025)
- [Scalable anomaly detection via a univariate Christoffel function](https://mlsanity.com/paper/2606.12483/) (2026)

## Built on by (most cited first, 1 of 1)

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

## Abstract (arXiv)

Semi-supervised anomaly detection plays a key role in diverse fields such as process monitoring, healthcare, and finance. However, lightweight methods often struggle with high-dimensional data and typically require careful tuning of multiple hyperparameters. Among existing approaches, Christoffel Function--based methods are attractive due to their simplicity, requiring at most a single hyperparameter. They also benefit from a well-established theoretical foundation that yields several interesting results for data science. However, their main limitation is poor scalability to high-dimensional settings. In this paper, we introduce CLOE, a new method that combines an autoencoder for dimensionality reduction with a Christoffel Function--based detector applied in the latent space. To better align representation learning with anomaly detection, we design a novel loss function that leverages the Christoffel Function to guide the autoencoder toward representations that better capture the support of the normal data distribution. We further propose a principled procedure to set the detection threshold and an efficient strategy to tune the single remaining hyperparameter. Experiments on multiple high-dimensional tabular anomaly detection benchmarks demonstrate that CLOE achieves superior performance compared to existing methods, while preserving the lightweight and low-tuning advantages of Christoffel Function--based approaches.