Time Series Classification through Diffeomorphic Time Warping (DiffTW)

2026-06-22

AI Summary

This study addresses the challenge of inaccurate similarity measurement in time series classification caused by nonlinear temporal deformations. To overcome the limitations of traditional dynamic time warping (DTW), which relies solely on discrete alignments, this work proposes the first continuous alignment framework based on diffeomorphic flows and optimal control theory. The method learns a continuous, invertible mapping between two real-valued functions by solving an ordinary differential equation linked to the characteristic curves of a linear transport equation. The velocity field governing the deformation is parameterized within a reproducing kernel Hilbert space, and the objective function is formulated using principles from optimal control. Evaluated via 1-nearest-neighbor classification across 86 benchmark datasets, the proposed approach outperforms DTW on 60 datasets, achieving significantly higher classification accuracy.

Abstract

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to a discrete response variable, like class labels. This task is fundamental in domains, including health monitoring, where the temporal structure of data is critical for accurate prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. To move beyond pairwise alignment, we propose a theoretical framework that learns mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between two time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function used to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms DTW on 60 of 86 datasets.

Problem

• Time Series Classification
• Dynamic Time Warping
• Diffeomorphic Transformation
• Temporal Alignment
• Similarity Measurement

Innovation

• Diffeomorphic Time Warping
• Dynamic Time Warping
• Time Series Classification
• Optimal Control
• Reproducing Kernel Hilbert Space