Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces
Julian Richter, Christopher Andrew Erdös, Christian Scheurer, Jochen J. Steil, Niels Dehio
AI summary
Problem
Existing time warping methods are largely restricted to Euclidean space or limited to aligning only two signals, leaving a gap for efficiently aligning multiple sequential signals that naturally reside on curved Riemannian manifolds.
Approach
The method projects signals into local tangent spaces, uses windowed sinc interpolation to model continuous time warping, and applies gradient-based optimization to align multiple sequences while preserving the underlying manifold geometry.
Key results
- Linear O(NZ) computational complexity for aligning multiple sequences
- Consistently outperforms TTW and NTW in signal averaging and classification benchmarks
- Successfully validated on synthetic data, the UCR time series archive, and real-robot motion learning
- First generic time warping framework applicable to arbitrary Riemannian manifolds beyond unit quaternions
Why it matters
Enables accurate temporal alignment for robotics, motion learning, and signal processing tasks where data inherently lies on non-Euclidean geometric structures.
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 at- tempt 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.