Persistent homology is an algebraic topological tool developed for the analysis of spatial data. It measures changes in topology of a filtration: a growing sequence of spaces indexed by a single real parameter. Persistent homology provides invariants called the barcodes or persistence diagrams that are sets of intervals recording the birth and death parameter values of each homology class in the filtration. Statistical analysis of persistent homology has been difficult because the raw information (the persistence diagrams) are provided as sets of intervals rather than functions. Many research groups are pursuing various approaches to converting persistence diagrams to functional forms that are then amenable to standard statistical techniques. One possibility is based on the persistent rank functions which are analogous to 2D cumulative sums of persistence intervals. A successful application of this technique is the analysis of experimentally imaged configurations of approximately mono-disperse spherical bead packings. The persistence diagrams highlight the regular tetrahedral and regular octahedral configurations of crystalline sphere packing in an unambiguous way and have led to new insights in the grain-scale mechanisms underlying the order-disorder transition in this dissipative, athermal system. Functional principle component analysis of the rank functions also reveals that there is effectively a single axis of variation in the experimental data samples.
Persistent homology of spatially random systems
Vanessa Robbins (ANU, Canberra)
Wed, 04/10/2017 - 09:00
Queens Building, room E303