  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.
