  Data depth is a concept in multivariate statistics that measures the
centrality of a point in a given data cloud in $\IR^d$. 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 in literature. 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 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.
