# Computing Topological Persistence for Simplicial Maps with Application to Data Sparsification

Friday, May 2, 2014 - 9:00am - 9:50am

Keller 3-180

Tamal Dey (The Ohio State University)

Recent data sparsification strategies in topological data analysis such as Graph Induced Complex and sparsified Rips complex give rise to a sequence of simplicial complexes connected by simplicial maps rather than inclusions. As a result, the need for computing topological persistence under such maps arises. We propose a practical algorithm for computing such persistence in Z2-homology.

A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. We exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. We exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

MSC Code:

46B85

Keywords: