  The regression depth of a hyperplane with respect to a set of n points in R^d
is the minimum number of points the hyperplane must pass through in a rotation
to vertical. We generalize hyperplane regression depth to k-flats for any k
between 0 and d-1. The k=0 case gives the classical notion of center points. We
prove that for any k and d, deep k-flats exist, that is, for any set of n
points there always exists a k-flat with depth at least a constant fraction of
n. As a consequence, we derive a linear-time (1+epsilon)-approximation
algorithm for the deepest flat.
