Extension of semi-Lipschitz maps on non-subadditive quasi-metric spaces: new tools for artificial intelligence

Abstract

The axioms of symmetry and indiscernibility of a metric function can be eliminated and still define a topological space with good enough  properties. However, among all the axioms that constitute the notion of a metric, the subadditivity or triangular inequality is the one that  has been more difficult to relax. Even so, and due to the potential applications in other fields such as Artificial Intelligence, several efforts  have been made in this direction. The notion of quasi-Banach space or the notion of b-metric space, are examples of structures where the  subadditivity has been replaced by a weaker inequality. Following this line, in this paper we consider the non-symmetric version of  the so called strong b-metric spaces. We focus our attention on the possibility of extending semi-Lipschitz maps, and still preserve (a uniform multiple of) the Lipschitz constant. As our main result, we prove that the possibility of obtaining such extensions characterizes  the spaces satisfying the proposed weak versions of the triangular inequality. We also show how our results can be applied in Machine  Learning to solve some technical issues, in particular, some problems related to the extension of Lipschitz functions with respect to the   cosine distance.