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 of 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.

Fairness in data mining tasks like clustering has recently become an increasingly important aspect. However, few clustering algorithms exist that focus on fair groupings of data with sensitive attributes. Including fairness in the clustering objective is especially hard for density-based clustering, as it does not directly optimize a closed form objective like centroid-based or spectral methods. This paper introduces FairDen, the first fair, density-based clustering algorithm. We capture the dataset's density-connectivity structure in a similarity matrix that we manipulate to encourage a balanced clustering. In contrast to state-of-the-art, FairDen inherently handles categorical attributes, noise, and data with several sensitive attributes or groups. We show that FairDen finds meaningful and fair clusters in extensive experiments.