# DCFO: Density-Based Counterfactuals for Outliers -- Additional Material

arXiv: https://arxiv.org/abs/2512.10659 · HTML: https://mlsanity.com/paper/2512.10659/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 13 of 13)

- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Opportunities and Challenges in Explainable Artificial Intelligence (XAI): A Survey](https://mlsanity.com/paper/2006.11371/) (2020)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Counterfactual Explanations and Algorithmic Recourses for Machine Learning: A Review](https://mlsanity.com/paper/2010.10596/) (2020)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [Diffusion Models for Counterfactual Generation and Anomaly Detection in Brain Images](https://mlsanity.com/paper/2308.02062/) (2023)
- [Diverse, Global and Amortised Counterfactual Explanations for Uncertainty Estimates](https://mlsanity.com/paper/2112.02646/) (2021)
- [Framing Algorithmic Recourse for Anomaly Detection](https://mlsanity.com/paper/2206.14384/) (2022)
- [Anomaly Attribution of Multivariate Time Series using Counterfactual Reasoning](https://mlsanity.com/paper/2109.06562/) (2021)
- [Show Me How: Benefits and Challenges of Agent-Augmented Counterfactual Explanations for Non-Expert Users](https://mlsanity.com/paper/2504.13897/) (2025)
- [AR-Pro: Counterfactual Explanations for Anomaly Repair with Formal Properties](https://mlsanity.com/paper/2410.24178/) (2024)

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

- [ExDBSCAN: Explaining DBSCAN with Counterfactual Reasoning -- Additional Material](https://mlsanity.com/paper/2605.30225/) (2026)

## Abstract (arXiv)

Outlier detection identifies data points that significantly deviate from the majority of the data distribution. Explaining outliers is crucial for understanding the underlying factors that contribute to their detection, validating their significance, and identifying potential biases or errors. Effective explanations provide actionable insights, facilitating preventive measures to avoid similar outliers in the future. Counterfactual explanations clarify why specific data points are classified as outliers by identifying minimal changes required to alter their prediction. Although valuable, most existing counterfactual explanation methods overlook the unique challenges posed by outlier detection, and fail to target classical, widely adopted outlier detection algorithms. Local Outlier Factor (LOF) is one the most popular unsupervised outlier detection methods, quantifying outlierness through relative local density. Despite LOF's widespread use across diverse applications, it lacks interpretability. To address this limitation, we introduce Density-based Counterfactuals for Outliers (DCFO), a novel method specifically designed to generate counterfactual explanations for LOF. DCFO partitions the data space into regions where LOF behaves smoothly, enabling efficient gradient-based optimisation. Extensive experimental validation on 50 OpenML datasets demonstrates that DCFO consistently outperforms benchmarked competitors, offering superior proximity and validity of generated counterfactuals.