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

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Abstract
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. We utilize the Banach space structure to define bounded variation paths of persistence diagrams, which can be studied using the path signature, a reparametrization-invariant characterization of paths valued in a Banach space. The signature is universal and characteristic which allows us to theoretically characterize measures on the space of paths and motivates its use in the context of kernel methods. However, kernel methods often require a feature map into a Hilbert space, so we introduce the moment map, a stable and injective feature map for static persistence diagrams, and compose it with the discrete path signature, producing a computable feature map into a Hilbert space. Finally, we demonstrate the efficacy of our methods by applying this to a parameter estimation problem for a 3D model of swarming behavior.

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