On the geometry of multivariate L ₁ objective functions

Summary:

Two multivariate L ₁ objective functions, namely the k–variate extensions of the classical mean deviation and mean difference, are considered. The duality between the original data vectors and the hyperplanes going through the origin and k – 1 data points is discussed and, consequently, different interesting representations and interpretations of the multivariate mean deviation are introduced. A similar duality is found between the lift data vectors and the hyperplanes going through k data points leading to different representations of the multivariate mean difference. The objective functions are also shown to have interpretations in terms of the centers of facets of the data based zonotopes and lift zonotopes. Moreover, interchanging the roles of the data vectors and the data hyperplanes yields nonparametric measures of (angular) distances between the data vectors as well as between the hyperplanes. Finally, multivariate sign and rank based tests and estimates in the one–sample and several–samples multivariate cases are discussed to illustrate the theory.