A topological approach to mapping space signatures
A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0,1]→Rn and Φ is the path signature...
| Main Authors: | , , , |
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| Format: | Internet publication |
| Language: | English |
| Published: |
2022
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| _version_ | 1797109831850524672 |
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| author | Giusti, C Lee, D Nanda, V Oberhauser, H |
| author_facet | Giusti, C Lee, D Nanda, V Oberhauser, H |
| author_sort | Giusti, C |
| collection | OXFORD |
| description | A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0,1]→Rn and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0,1]d→Rn for any d∈N, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d≥2. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces. |
| first_indexed | 2024-03-07T07:45:18Z |
| format | Internet publication |
| id | oxford-uuid:52d8730e-315e-40b4-9869-ccd5c6973c23 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:45:18Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | oxford-uuid:52d8730e-315e-40b4-9869-ccd5c6973c232023-06-09T09:18:44ZA topological approach to mapping space signaturesInternet publicationhttp://purl.org/coar/resource_type/c_7ad9uuid:52d8730e-315e-40b4-9869-ccd5c6973c23EnglishSymplectic Elements2022Giusti, CLee, DNanda, VOberhauser, HA common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0,1]→Rn and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0,1]d→Rn for any d∈N, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d≥2. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces. |
| spellingShingle | Giusti, C Lee, D Nanda, V Oberhauser, H A topological approach to mapping space signatures |
| title | A topological approach to mapping space signatures |
| title_full | A topological approach to mapping space signatures |
| title_fullStr | A topological approach to mapping space signatures |
| title_full_unstemmed | A topological approach to mapping space signatures |
| title_short | A topological approach to mapping space signatures |
| title_sort | topological approach to mapping space signatures |
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