Exact weights, path metrics, and algebraic Wasserstein distances
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866913694108090368 |
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| author | Bubenik, Peter Scott, Jonathan Stanley, Donald |
| author_facet | Bubenik, Peter Scott, Jonathan Stanley, Donald |
| contents | We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_09654 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Exact weights, path metrics, and algebraic Wasserstein distances Bubenik, Peter Scott, Jonathan Stanley, Donald Rings and Algebras Algebraic Topology We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms. |
| title | Exact weights, path metrics, and algebraic Wasserstein distances |
| topic | Rings and Algebras Algebraic Topology |
| url | https://arxiv.org/abs/1809.09654 |