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

TL;DR

This paper introduces a new class of metrics applicable to sets, vectors, functions, and structured data, unifying and extending existing metrics, with proven properties and demonstrated effectiveness in data mining tasks.

Contribution

The paper presents a novel, unified class of metrics that generalize popular distances and are suitable for high-dimensional and structured data analysis.

Findings

New metrics are complete and relate to $f$-divergences.

Empirical results show improved performance over traditional metrics.

Metrics are effective for high-dimensional and structured data processing.

Abstract

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 concept hierarchies and ontologies, we introduce information-theoretic metrics on directed acyclic graphs drawn according to a fixed probability distribution. We conduct empirical investigation to demonstrate intuitive interpretation of the new metrics and their effectiveness on real-valued, high-dimensional, and structured data. Extensive comparative evaluation demonstrates that the new metrics outperformed multiple similarity and dissimilarity functions traditionally used in data mining, including the Minkowski family, the fractional $L^p$ family, two $f$-divergences, cosine distance, and two correlation coefficients. Finally, we argue that the new class of metrics is particularly appropriate for rapid processing of high-dimensional and structured data in distance-based learning.