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Data depth is a statistical method which models data distribution in terms of center-outward ranking rather than density or linear ranking. While there are a lot of academic interests, its applications are hampered by the lack of a method which is both robust and efficient. This paper introduces
Half-Space Mass
which is a significantly improved version of half-space data depth.
Half-Space Mass
is the only data depth method which is both robust and efficient, as far as we know. We also reveal four theoretical properties of
Half-Space Mass
: (i) its resultant mass distribution is concave regardless of the underlying density distribution, (ii) its maximum point is unique which can be considered as median, (iii) the median is maximally robust, and (iv) its estimation extends to a higher dimensional space in which the convex hull of the dataset occupies zero volume. We demonstrate the power of
Half-Space Mass
through its applications in two tasks. In anomaly detection, being a maximally robust location estimator leads directly to a robust anomaly detector that yields a better detection accuracy than half-space depth; and it runs orders of magnitude faster than
L
2
depth, an existing maximally robust location estimator. In clustering, the
Half-Space Mass
version of K-means overcomes three weaknesses of K-means.