Persistent Intersection Homology for the Analysis of Discrete Data
This work addresses a specific challenge in topological data analysis for researchers dealing with complex, multi-manifold data sets, representing an incremental extension of existing methods.
The paper tackles the problem of analyzing discrete data sets that represent multiple manifolds with different dimensions, which existing persistent homology methods cannot handle, by introducing persistent intersection homology to extract features and manage singularities in visualization.
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.