L1-Depth Revisited: A Robust Angle-Based Outlier Factor in High-Dimensional Space

  • Ninh PhamEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11051)


Angle-based outlier detection (ABOD) has been recently emerged as an effective method to detect outliers in high dimensions. Instead of examining neighborhoods as proximity-based concepts, ABOD assesses the broadness of angle spectrum of a point as an outlier factor. Despite being a parameter-free and robust measure in high-dimensional space, the exact solution of ABOD suffers from the cubic cost \(O(n^3)\) regarding the data size n, hence cannot be used on large-scale data sets.

In this work we present a conceptual relationship between the ABOD intuition and the L1-depth concept in statistics, one of the earliest methods used for detecting outliers. Deriving from this relationship, we propose to use L1-depth as a variant of angle-based outlier factors, since it only requires a quadratic computational time as proximity-based outlier factors. Empirically, L1-depth is competitive (often superior) to proximity-based and other proposed angle-based outlier factors on detecting high-dimensional outliers regarding both efficiency and accuracy.

In order to avoid the quadratic computational time, we introduce a simple but efficient sampling method named SamDepth for estimating L1-depth measure. We also present theoretical analysis to guarantee the reliability of SamDepth. The empirical experiments on many real-world high-dimensional data sets demonstrate that SamDepth with \(\sqrt{n}\) samples often achieves very competitive accuracy and runs several orders of magnitude faster than other proximity-based and ABOD competitors. Data related to this paper are available at: Code related to this paper is available at:



We would like to thank Rasmus Pagh for useful discussion and comments in the early stage of this work. We thank members of the DABAI project and anonymous reviewers for their constructive comments and suggestions.

Supplementary material

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Supplementary material 1 (pdf 159 KB)


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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

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