  Path signatures are powerful nonparametric tools for time series analysis,
shown to form a universal and characteristic feature map for Euclidean valued
time series data. We lift the theory of path signatures to the setting of Lie
group valued time series, adapting these tools for time series with underlying
geometric constraints. We prove that this generalized path signature is
universal and characteristic. To demonstrate universality, we analyze the human
action recognition problem in computer vision, using $SO(3)$ representations
for the time series, providing comparable performance to other shallow learning
approaches, while offering an easily interpretable feature set. We also provide
a two-sample hypothesis test for Lie group-valued random walks to illustrate
its characteristic property. Finally we provide algorithms and a Julia
implementation of these methods.
