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Multivariate Regression Depth

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.