Mapping Spaces, Signatures, And Data
Time-varying phenomena are ubiquitous across pure and applied mathematics, from path spaces and stochastic differential equations to multivariate time series and dynamic point clouds. The path signature provides a powerful characterization of such sequential data in terms of power series of tensors, weaving together these diverse concepts. Originally defined as part of Chen's iterated integral cochain algebra, the path signature has since been used as the foundation for the theory of rough paths in stochastic analysis. More recently, it has been shown to be a universal and characteristic feature map for multivariate time series, providing theoretical guarantees for its application to time series analysis in the context of kernel methods in machine learning. This thesis extends the scope of the path signature to more complex parametrized data in two directions. First, we consider generalizations of the codomain of a path. We lift the theory of signatures to the setting of Lie group valued time series, adapting these tools for time series with underlying geometric constraints. Furthermore, we build a signature framework to study paths of persistence diagrams, objects which capture the evolving topological structure of dynamic data sets. Second, we consider maps parametrized by higher dimensional cubes by developing notions of the mapping space signature. Our approach returns to the topological origins of the signature as the 0-cochains of the Chen construction. We formulate a cubical variant of the mapping space construction, and use the resulting 0-cochains to define the mapping space signature and establish its basic properties.