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Topology Applied to Machine Learning: From Global to Local

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural im...

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Autores principales: Adams, Henry, Moy, Michael
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Frontiers Media S.A. 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8160457/
https://www.ncbi.nlm.nih.gov/pubmed/34056580
http://dx.doi.org/10.3389/frai.2021.668302
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author Adams, Henry
Moy, Michael
author_facet Adams, Henry
Moy, Michael
author_sort Adams, Henry
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description Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.
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spelling pubmed-81604572021-05-29 Topology Applied to Machine Learning: From Global to Local Adams, Henry Moy, Michael Front Artif Intell Artificial Intelligence Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning. Frontiers Media S.A. 2021-05-14 /pmc/articles/PMC8160457/ /pubmed/34056580 http://dx.doi.org/10.3389/frai.2021.668302 Text en Copyright © 2021 Adams and Moy. https://creativecommons.org/licenses/by/4.0/This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
spellingShingle Artificial Intelligence
Adams, Henry
Moy, Michael
Topology Applied to Machine Learning: From Global to Local
title Topology Applied to Machine Learning: From Global to Local
title_full Topology Applied to Machine Learning: From Global to Local
title_fullStr Topology Applied to Machine Learning: From Global to Local
title_full_unstemmed Topology Applied to Machine Learning: From Global to Local
title_short Topology Applied to Machine Learning: From Global to Local
title_sort topology applied to machine learning: from global to local
topic Artificial Intelligence
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8160457/
https://www.ncbi.nlm.nih.gov/pubmed/34056580
http://dx.doi.org/10.3389/frai.2021.668302
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