Geodesic
A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison
Applications of Mathematics, Tome 68 (2023) no. 2, pp. 143-152.

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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.
DOI : 10.21136/AM.2021.0090-21
Classification : 46A40, 94A08
Keywords: generalized metric; vector-valued metric; scalarization; image comparison; structural similarity index 
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Piella, Gemma. A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison. Applications of Mathematics, Tome 68 (2023) no. 2, pp. 143-152. doi : 10.21136/AM.2021.0090-21. http://geodesic.mathdoc.fr/articles/10.21136/AM.2021.0090-21/

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