Path Signatures on Lie Groups
This enables time series analysis for data with geometric constraints, such as in computer vision, though it is incremental as it adapts existing tools to a new setting.
The paper extends path signatures to Lie group-valued time series, proving they are universal and characteristic features, and demonstrates this by achieving comparable performance in human action recognition using SO(3) representations.
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.