Pseudo-Multidimensional Persistence and Its Applications

While one-dimensional persistent homology can be an effective way to discriminate data, it has limitations. Multidimensional persistent homology is a technique amenable to data naturally described by more than a single parameter, and is able to encoding more robust information about the structure of the data. However, as indicated by Carlsson and Zomorodian (Discrete Comput Geom 42(1):71–271, 2009), no perfect higher-dimensional analogue of the one-dimensional persistence barcode exists for higher-dimensional filtrations. Xia and Wei (J Comput Chem 36:1502–1520, 2015) propose computing one-dimensional Betti number functions at various values of a second parameter and stacking these functions for each homological dimension. The aim of this visualization is to increase the discriminatory power of current one-dimensional persistence techniques, especially for datasets that have features more readily captured by a combination of two parameters. We apply this practical approach to three datasets, relating to (1) craniofacial shape and (2) Lissajous knots, both using parameters for scale and curvature; and (3) the Kuramoto–Sivashinsky partial differential equation, using parameters for both scale and time. This new approach is able to differentiate between topologically equivalent geometric objects and offers insight into the study of the Kuramoto–Sivashinsky partial differential equation and Lissajous knots. We were unable to obtain meaningful results, however, in our applications to the screening of anomalous facial structures, although our method seems sensitive enough to identify patients at severe risk of a sleep disorder associated closely with craniofacial structure. This approach, though still in its infancy, presents new insights and avenues for the analysis of data with complex structure.