On Aggregating Two Metrics with Relaxed Triangle Inequalities by the Weighted Harmonic Mean

Toshiya ITOH  Yoshinori TAKEI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A   No.9   pp.1404-1411
Publication Date: 2018/09/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.1404
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
aggregation of metrics,  relaxed triangle inequality,  set similarity,  Jaccard-Tanimoto distance,  

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An important problem in mathematics and data science, given two or more metric spaces, is obtaining a metric of the product space by aggregating the source metrics using a multivariate function. In 1981, Borsík and Doboš solved the problem, and much progress has subsequently been made in generalizations of the problem. The triangle inequality is a key property for a bivariate function to be a metric. In the metric aggregation, requesting the triangle inequality of the resulting metric imposes the subadditivity on the aggregating function. However, in some applications, such as the image matching, a relaxed notion of the triangle inequality is useful and this relaxation may enlarge the scope of the aggregators to include some natural superadditive functions such as the harmonic mean. This paper examines the aggregation of two semimetrics (i.e. metrics with a relaxed triangle inequality) by the harmonic mean is studied and shows that such aggregation weakly preserves the relaxed triangle inequalities. As an application, the paper presents an alternative simple proof of the relaxed triangle inequality satisfied by the robust Jaccard-Tanimoto set dissimilarity, which was originally shown by Gragera and Suppakitpaisarn in 2016.