The mixed Lipschitz space and its dual for tree metrics

Abstract

This paper develops a theory of harmonic analysis on spaces with tree metrics, extending previous work in this direction by Gavish, Nadler and Coifman (2010) [30] and Gavish and Coifman (2011, 2012) [28] , [29] . We show how a natural system of martingales and martingale differences induced by a partition tree leads to simple and effective characterizations of the Lipschitz norm and its dual for functions on a single tree metric space. The restrictions we place on the tree metrics are far more general than those considered in previous work. As the dual norm is equal to the Earth Mover's Distance (EMD) between two probability distributions, we recover a simple formula for EMD with respect to tree distances presented by Charikar (2002) [36] . We also consider the situation where an arbitrary metric is approximated by the average of a family of dominating tree metrics. We show that the Lipschitz norm and its dual for the tree metrics can be combined to yield an approximation to the corresponding norms for the underlying metric. The main contributions of this paper, however, are the generalizations of the aforementioned results to the setting of the product of two or more tree metric spaces. For functions on a product space, the notion of regularity we consider is not the Lipschitz condition, but rather the mixed Lipschitz condition that controls the size of a function's mixed difference quotient. This condition is extremely natural for datasets that can be described as a product of metric spaces, such as word-document databases. We develop effective formulas for norms equivalent to the mixed Lipschitz norm and its dual, and extend our results on combining pairs of trees.