  Homology-based invariants can be used to characterize the geometry of
datasets and thereby gain some understanding of the processes generating those
datasets. In this work we investigate how the geometry of a dataset changes
when it is subsampled in various ways. In our framework the dataset serves as a
reference object; we then consider different points in the ambient space and
endow them with a geometry defined in relation to the reference object, for
instance by subsampling the dataset proportionally to the distance between its
elements and the point under consideration. We illustrate how this process can
be used to extract rich geometrical information, allowing for example to
classify points coming from different data distributions.
