IEICE TRANSACTIONS on Fundamentals
On Aggregating Two Metrics with Relaxed Triangle Inequalities by the Weighted Harmonic Mean
Summary :
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.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E101-A No.9 pp.1404-1411
- Publication Date
- 2018/09/01
- Publicized
- 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)
- Category
Authors
Toshiya ITOH
Tokyo Institute of Technology
Yoshinori TAKEI
Akita College
Keyword
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Toshiya ITOH, Yoshinori TAKEI, "On Aggregating Two Metrics with Relaxed Triangle Inequalities by the Weighted Harmonic Mean" in IEICE TRANSACTIONS on Fundamentals,
vol. E101-A, no. 9, pp. 1404-1411, September 2018, doi: 10.1587/transfun.E101.A.1404.
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.1404/_p
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@ARTICLE{e101-a_9_1404,
author={Toshiya ITOH, Yoshinori TAKEI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On Aggregating Two Metrics with Relaxed Triangle Inequalities by the Weighted Harmonic Mean},
year={2018},
volume={E101-A},
number={9},
pages={1404-1411},
abstract={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.},
keywords={},
doi={10.1587/transfun.E101.A.1404},
ISSN={1745-1337},
month={September},}
Copy
TY - JOUR
TI - On Aggregating Two Metrics with Relaxed Triangle Inequalities by the Weighted Harmonic Mean
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1404
EP - 1411
AU - Toshiya ITOH
AU - Yoshinori TAKEI
PY - 2018
DO - 10.1587/transfun.E101.A.1404
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2018
AB - 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.
ER -
English
