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. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

翻译：我们引入了 $ varepsilon\ geq 0$ 的 Euclidean 矢量捆绑概念的$varepsilon$- 近似版本。 美元 varepsilon\ geq 0$ 将恢复了 Euclidean 矢量捆绑的传统概念, 当美元 varepsilon = 0$ 美元 时, 我们特别用一定数量的数据对具有满足大约共同循环条件的正方圆组中的系数的矢量捆绑进行了研究。 我们显示, 美元 varepsilon $- apblobal 矢量捆绑可以用来代表典型的矢量捆绑。 我们还在大约的矢量捆绑之间引入距离, 并用它们来证明足够相似的矢量矢量的矢量捆绑是相同的典型。 这样可以使用一定数量的数据, 并允许在与矢量捆绑的矢量一起工作时对噪音有某种容忍度。 例如, 我们证明从有限的样品中重建了矢量捆绑的矢量。 我们用算算法来有效计算低维度的矢量特性特性特性, 用离离离的矢量和运的模型的计算。