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

Gemma Piella

Applications of Mathematics (2023)

  • Volume: 68, Issue: 2, page 143-152
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Piella, Gemma. "A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison." Applications of Mathematics 68.2 (2023): 143-152. <http://eudml.org/doc/299515>.

@article{Piella2023,
abstract = {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.},
author = {Piella, Gemma},
journal = {Applications of Mathematics},
keywords = {generalized metric; vector-valued metric; scalarization; image comparison; structural similarity index},
language = {eng},
number = {2},
pages = {143-152},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison},
url = {http://eudml.org/doc/299515},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Piella, Gemma
TI - A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 143
EP - 152
AB - 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.
LA - eng
KW - generalized metric; vector-valued metric; scalarization; image comparison; structural similarity index
UR - http://eudml.org/doc/299515
ER -

References

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