Persistence Images: A Stable Vector Representation of Persistent\n Homology

Many datasets can be viewed as a noisy sampling of an underlying space, and\ntools from topological data analysis can characterize this structure for the\npurpose of knowledge discovery. One such tool is persistent homology, which\nprovides a multiscale description of the homological features within a dataset.\nA useful representation of this homological information is a persistence\ndiagram (PD). Efforts have been made to map PDs into spaces with additional\nstructure valuable to machine learning tasks. We convert a PD to a\nfinite-dimensional vector representation which we call a persistence image\n(PI), and prove the stability of this transformation with respect to small\nperturbations in the inputs. The discriminatory power of PIs is compared\nagainst existing methods, showing significant performance gains. We explore the\nuse of PIs with vector-based machine learning tools, such as linear sparse\nsupport vector machines, which identify features containing discriminating\ntopological information. Finally, high accuracy inference of parameter values\nfrom the dynamic output of a discrete dynamical system (the linked twist map)\nand a partial differential equation (the anisotropic Kuramoto-Sivashinsky\nequation) provide a novel application of the discriminatory power of PIs.\n