A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison

Distance metrics are at the core of many processing and machine learning algorithms. In many contexts, it is useful to compute the distance between data using multiple criteria. This naturally leads to consider vector-valued metrics, in which the distance is no longer a real positive number but a ve...

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Bibliographic Details
Published inApplications of mathematics (Prague) Vol. 68; no. 2; pp. 143 - 152
Main Author Piella, Gemma
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2022
Springer Nature B.V
Subjects
Algorithms
Analysis
Applications of Mathematics
Classical and Continuum Physics
Machine learning
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Multiple criterion
Optimization
Theoretical
94A08
image comparison
generalized metric
scalarization
vector-valued metric
structural similarity index
46A40
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Summary:Distance metrics are at the core of many processing and machine learning algorithms. In many contexts, it is useful to compute the distance between data using multiple criteria. This naturally leads to consider vector-valued metrics, in which the distance is no longer a real positive number but a vector. In this paper, we propose a principled way to combine several metrics into either a scalar-valued or vector-valued metric. We illustrate our framework by reformulating the popular structural similarity (SSIM) index and a simple case of the Wasserstein distance used for optimal transport.
ISSN:0862-7940
1572-9109
DOI:10.21136/AM.2021.0090-21