  Topological data analysis is becoming increasingly relevant to support the
analysis of unstructured data sets. A common assumption in data analysis is
that the data set is a sample---not necessarily a uniform one---of some
high-dimensional manifold. In such cases, persistent homology can be
successfully employed to extract features, remove noise, and compare data sets.
The underlying problems in some application domains, however, turn out to
represent multiple manifolds with different dimensions. Algebraic topology
typically analyzes such problems using intersection homology, an extension of
homology that is capable of handling configurations with singularities. In this
paper, we describe how the persistent variant of intersection homology can be
used to assist data analysis in visualization. We point out potential pitfalls
in approximating data sets with singularities and give strategies for resolving
them.
