Signatures, Lipschitz-free spaces, and paths of persistence diagrams

Understand

Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces.

• These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in multi-agent systems, time-varying fMRI scans from neuroscience, and time-dependent scalar fields in hydrodynamics.
• While persistence diagrams can provide a powerful topological summary of data, the standard space of persistence diagrams lacks the sufficient algebraic and analytic structure required for many theoretical and computational analyses.
• We enrich the space of persistence diagrams by isometrically embedding it into a Lipschitz-free space, a Banach space built from a universal construction.