Topological Inference from Measures

In this thesis, we study the problem of analysing topological structure in point cloud data. One
widely used tool in this domain is persistent homology. By processing the data at all scales, it
does not rely on a particular choice of scale, which is one of the main challenge faced in this
area. Moreover, its stability properties provide a natural connection between discrete data
and an underlying continuous structure. Finally, it can be combined with other tools, like the
distance to a measure, which allows to handle noise that are unbounded. The main caveat of
this approach is its high complexity.
In this thesis, we will introduce topological data analysis and persistent homology, then show
how to use approximation to reduce the computational complexity. We provide an approxima-
tion scheme to the distance to a measure and a sparsifying method of weighted Vietoris-Rips
complexes in order to approximate persistence diagrams with practical complexity. We detail
the specific properties of these constructions.
Persistent homology was previously shown to be of use for scalar field analysis. We provide
a way to combine it with the distance to a measure in order to handle a wider class of noise,
especially data with unbounded errors. Finally, we discuss interesting opportunities opened
by these results to study data where parts are missing or erroneous.