Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces

2025-06-02

AI Summary

Existing time-warping methods are primarily designed for Euclidean spaces and struggle to align multi-time-series signals embedded in Riemannian manifolds—such as robotic trajectories and orientations. This work introduces RTW, the first general-purpose Riemannian time-warping framework for multiple sequences. RTW explicitly incorporates manifold geometry by leveraging geodesic distances, dynamic programming, and manifold-constrained optimization, supporting canonical manifolds including the unit quaternion sphere and the space of symmetric positive-definite (SPD) matrices. Theoretically grounded and computationally efficient, RTW achieves significantly higher average alignment accuracy and downstream classification accuracy than state-of-the-art methods on both synthetic benchmarks and real-world KUKA LBR iiwa robot motion data. By operating intrinsically on Riemannian manifolds, RTW overcomes the fundamental limitation of conventional time-warping approaches, which are restricted to Euclidean domains.

Technology Category

• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Temporal alignment of multiple signals through time warping is crucial in many fields, such as classification within speech recognition or robot motion learning. Almost all related works are limited to data in Euclidean space. Although an attempt was made in 2011 to adapt this concept to unit quaternions, a general extension to Riemannian manifolds remains absent. Given its importance for numerous applications in robotics and beyond, we introduce Riemannian Time Warping~(RTW). This novel approach efficiently aligns multiple signals by considering the geometric structure of the Riemannian manifold in which the data is embedded. Extensive experiments on synthetic and real-world data, including tests with an LBR iiwa robot, demonstrate that RTW consistently outperforms state-of-the-art baselines in both averaging and classification tasks.

Problem

• Extends time warping to Riemannian manifolds
• Aligns multiple signals in curved spaces
• Improves averaging and classification accuracy

Innovation

• Extends time warping to Riemannian manifolds
• Aligns signals using geometric manifold structure
• Outperforms baselines in averaging and classification