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Title: Approximate computation of projection depths Authors:  Rainer Dyckerhoff - University of Cologne (Germany)
Pavlo Mozharovskyi - Telecom Paris, Institut Polytechnique de Paris (France) [presenting]
Stanislav Nagy - Charles University (Czech Republic)
Abstract: Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in a Euclidean space. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed. For depths that satisfy the projection property, an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as this is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method, we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.