Weighted-Digraph-Guided Multi-Kernelized Learning for Outlier Explanation

Outlier explanation methods based on outlying subspace mining have been widely used in various applications due to their effectiveness and explainability. These existing methods aim to find an outlying subspace of the original space (a set of features) that can clearly distinguish a query outlier from all inliers. However, when the query outlier in the original space are linearly inseparable from inliers, these existing methods may not be able to accurately identify an outlying subspace that effectively distinguishes the query outlier from all inliers. Moreover, these methods ignore differences between the query outlier and other outliers. In this paper, we propose a novel method named WANDER (Wighted-digrAph-Guided Multi-KerNelizeD lEaRning) for outlier explanation, aiming to learn an optimal outlying subspace that can separate the query outlier from other outliers and the inliers simultaneously. Specifically, we first design a quadruplet sampling module to transform the original dataset into a set of quadruplets to mitigate extreme data imbalances and to help the explainer better capture the differences among the query outlier, other outliers, and inliers. Then we design a weighted digraph generation module to capture the geometric structure in each quadruplet within the original space. In order to consider the condition that quadruplets are linearly inseparable in the original space, we further construct a feature embedding module to map the set of quadruplets from the original space to a kernelized embedding space. To find the optimal kernelized embedding space, we design an outlying measure module to iteratively update the parameters in the feature embedding module by the weighted-digraph-based quadruplet loss. Finally, WANDER outputs an outlying subspace used to interpret the query outlier through an outlying subspace extraction module. Extensive experiments show that WANDER outperforms state-of-the-art methods, achieving improvements in AUPRC, AUROC, Jaccard Index, and F 1 scores of up to 25.3%, 16.5%, 37.4%, and 28.4%, respectively, across seven real-world datasets. Our datasets and source code are publicly available at https://github.com/KDDElab/WANDER1.