Path Signatures on Lie Groups

Abstract: 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.

• The paper introduces a novel extension of classical path signatures from Euclidean spaces to Lie groups, providing a universal and characteristic feature map.
• It employs iterated integrals over Lie algebras and offers a Julia implementation, ensuring efficient computation for complex time series analysis.
• The approach demonstrates practical advantages in human action recognition and two-sample hypothesis testing by accurately capturing rotational dynamics.

Summary of "Path Signatures on Lie Groups"

Introduction

The paper "Path Signatures on Lie Groups" (2007.06633) expands the theory of path signatures — a robust nonparametric tool traditionally used for time series analysis in Euclidean spaces — to the complex setting of Lie groups. This adaptation is crucial for handling time series data with intrinsic geometric constraints, as seen in applications involving rotation and motion in spaces like $SO(3)$. The authors demonstrate that this generalized approach preserves the universality and characteristic properties observed in the Euclidean case, thus offering a feature map that is both theoretically sound and practically advantageous for tasks such as human action recognition.

Theoretical Foundation

Path Signature Definition

A Lie group is a manifold that supports a group structure, making it ideal for modeling phenomena with geometric constraints. The notion of path signatures is lifted to time series valued in Lie groups by employing iterated integrals over paths expressed through the Lie algebra, the tangent space at the group identity. This constructs a feature map within a tensor algebra that characterizes the path comprehensively up to tree-like equivalence, a quotient of paths exhibiting similar end-points and structural loops.

Key Features of Lie Group Path Signatures

• Universality and Characteristic Kernels: The generalized path signatures retain their ability to approximate any continuous bounded function over the path space, making them universal. Furthermore, they are characteristic as they can distinguish between probability measures on this space, enabling effective two-sample hypothesis testing.
• Computational Approach: The authors provide practical algorithms and a comprehensive Julia implementation for calculating path signatures in Lie groups. This extends the applicability of such computations beyond Euclidean settings to more complex motion and rotation spaces.

  Figure 1: An example of the function Phi when we take $G = S^1 \times S^1$. (Left) A path $\gamma \in PR^2_0$. (Middle) The derivative of $\gamma$ as a path in $r$ or $g$. (Right) The corresponding path $\Phi(\gamma)$.

Empirical Application

Human Action Recognition

The paper showcases the application of Lie group path signatures in human action recognition, employing $SO(3)$ representations of skeletal motions. By simplifying preprocessing — avoiding extensive processes like dynamic time warping — the authors achieve comparable results to other shallow learning methods with less complex data transformations. This highlights the efficiency and interpretability of path signatures, where second-level signature matrices provide insights into the underlying mechanics of human movement.

Figure 2: Averaged absolute second level signature matrix for the action class "walk."

Two-Sample Hypothesis Testing

Utilizing the characteristic property, the paper further elaborates on hypothesis tests distinguishing between distinct random walks on $SO(3)$. The Lie group path signatures notably outperform their Euclidean counterparts, demonstrating superior sensitivity to subtle differences in rotational drift, an invariant under group translations.

Figure 3: Test (top) and null (bottom) distributions of $MMD_u$ when $H_0$ is false.

Conclusion

The authors establish that path signatures on Lie groups retain both theoretical and practical strengths essential for modern time series challenges. This advancement opens avenues for analyzing data types constrained by geometric properties, such as rotations and motions, marking a significant stride in both theoretical development and expansive applicability across data science domains. Future research may explore manifold scenarios and explore further applications far beyond classical Euclidean settings.