Classical Multidimensional Scaling on Metric Measure Spaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913346918285312 |
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| author | Lim, Sunhyuk Memoli, Facundo |
| author_facet | Lim, Sunhyuk Memoli, Facundo |
| contents | We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^{d-1} and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_09385 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Classical Multidimensional Scaling on Metric Measure Spaces Lim, Sunhyuk Memoli, Facundo Functional Analysis Metric Geometry We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^{d-1} and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance. |
| title | Classical Multidimensional Scaling on Metric Measure Spaces |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2201.09385 |