Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction – how to assemble discrete samples into global structures, and inference – how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have shown that by constructing an offset of the samples using a suitable offset parameter could provide reconstructions that preserve homotopy types therefore homology for densely sampled smooth submanifolds of Euclidean space without boundary. Chazal et al. (2009) and Attali et al. (2013) have introduced a parameterized set of sampling conditions that extend the results of Niyogi et al. to a large class of compact subsets of Euclidean space. Our work tackles data problems that fill a gap between the work of Niyogi et al. and Chazal et al. In particular, we give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories. We also give stronger results that relate topological equivalence between the offset and the manifold as a deformation retract.

给定一组从某些基础空间中采样的数据点，在处理采样数据时，几何和拓扑数据分析面临两个重要挑战：重建-如何将离散样本组装成全局结构，以及推理-如何提取几何和拓扑信息来自高维，不完整和嘈杂的数据。Niyogi等。（2008年）表明，通过使用适当的偏移量参数构造样本的偏移量，可以提供保留同型性的重建，因此，对于无边界的欧氏空间密集采样的光滑子流形具有同源性。Chazal等。（2009年）和Attali等人。（2013年）引入了一组参数化的采样条件，以扩展Niyogi等人的结果。到欧几里得空间的一类紧致子集。我们的工作解决了填补Niyogi等人工作之间空白的数据问题。和Chazal等。特别是，我们给出了具有边界的流形采样条件的概率概念，而现有理论无法处理这种情况。我们还给出了更强的结果，这些结果将偏移量和歧管之间的拓扑等效性与变形回缩相关。