{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/approximate-and-discrete-euclidean-vector","title":"Approximate and discrete Euclidean vector bundles","arxiv_id":"2104.07563","date":"2021-04-15","proceeding":null,"authors":["Luis Scoccola","Jose A. Perea"],"abstract":"We introduce $\\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\\varepsilon \\geq 0$, which recover the classical notion of Euclidean vector bundle when $\\varepsilon = 0$. In particular, we study \\v{C}ech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\\varepsilon$-approximate vector bundles can be used to represent classical vector bundles when $\\varepsilon > 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. 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