Persistence Images: A Stable Vector Representation of Persistent Homology

Many data sets can be viewed as a noisy sampling of an underlying topological space. A suite of tools in topological data analysis allows one to exploit this structure for the purpose of knowledge discovery. One such tool is persistent homology which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). The space of PDs can be given a metric structure allowing a given diagram to be used as a statistic for the purpose of comparison against other diagrams. We convert a PD to a persistence image (PI) and prove stability with respect to small perturbations in the inputs. The PI is a vector representation allowing the application of vector-based machine learning tools, such as linear and sparse support vector machines. These tools help to identify discriminatory features which can have a topological interpretation. The PIs and PDs derived from randomly sampled topological spaces are compared by applying the K-medoids clustering algorithm. To further illustrate the PI technique, linear and sparse support vector machines are implemented on this data set and classification is performed on additional data arising from a discrete dynamical system called the linked twist map.