Time Series Classification through Diffeomorphic Time Warping (DiffTW)
2026-06-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to classify time-based sequences, like signals over time, into different categories. They improve on a common technique called Dynamic Time Warping (DTW), which aligns sequences by matching points, by creating a more advanced method that treats sequences as smooth functions and maps one to another using ideas from differential equations. Their approach models the transformation between sequences as flows influenced by changing speeds, making the comparison more flexible. Testing their method, called Diffeomorphic Time Warping (DiffTW), they found it worked better than DTW on many datasets.
time series classificationDynamic Time Warping (DTW)diffeomorphic transformationpartial differential equationsordinary differential equations (ODEs)method of characteristicsreproducing kernel Hilbert spacesoptimal control1-nearest neighbor classifier
Authors
Vicky Geneva Haney, Kamel Lahouel, Victor Rielly, Bruno M. Jedynak
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.