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$\varepsilon \geq 0$情形下欧氏向量丛概念的$\varepsilon$近似版本，当$\varepsilon=0$时该版本恢复为经典欧氏向量丛概念。特别地，我们研究了正交群系数的$\check{\text{C}}$ech上链满足近似上链条件的性质。结果表明，当$\varepsilon>0$足够小时，$\varepsilon$近似向量丛可用于表示经典向量丛。我们进一步引入近似向量丛间的距离，并利用该距离证明足够相似的近似向量丛能表示相同的经典向量丛。这为在有限单纯复形上利用有限数据指定向量丛提供了方法，同时也允许在应用场景处理向量丛时具有一定的噪声容限。作为实例，我们证明了从有限样本重建向量丛的定理。我们给出了直接从离散与近似表示有效计算向量丛低维示性类的算法，并通过计算实例展示了这些算法的应用。