A new class of metrics for learning on real-valued and structured data

We propose a new class of metrics on sets, vectors, and functions that can be used in various stages of data mining, including exploratory data analysis, learning, and result interpretation. These new distance functions unify and generalize some of the popular metrics, such as the Jaccard and bag distances on sets, Manhattan distance on vector spaces, and Marczewski-Steinhaus distance on integrable functions. We prove that the new metrics are complete and show useful relationships with f-divergences for probability distributions. To further extend our approach to structured objects such as ontologies, we introduce information-theoretic metrics on directed acyclic graphs drawn according to a fixed probability distribution. We conduct empirical investigation to demonstrate the effectiveness on real-valued, high-dimensional, and structured data. Overall, the new metrics compare favorably to multiple similarity and dissimilarity functions traditionally used in data mining, including the Minkowski (Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}) family, the fractional Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} family, two f-divergences, cosine distance, and two correlation coefficients. We provide evidence that they are particularly appropriate for rapid processing of high-dimensional and structured data in distance-based learning.