aiqbench analysis
This research introduces Diffeomorphic Time Warping (DiffTW), a theoretically grounded method that models time series alignment using partial differential equations and optimal control rather than discrete point matching. While the updated results show DiffTW performs comparably to unconstrained Dynamic Time Warping but lags behind constrained variants, it offers significant advantages in interpretability by explicitly modeling system dynamics via ordinary differential equations. The study highlights critical trade-offs between flexibility and performance constraints when applying diffeomorphic transformations to real-world datasets like those from UCR/UEA archives.
Time Series Classification through Diffeomorphic Time Warping (DiffTW)
arXiv:2606.23472v2 Announce Type: replace-cross Abstract: Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to discrete response variables, like class labels. This task is fundamental in domains, including health monitoring, where temporal structure is critical for 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. Moving beyond pairwise alignment, we propose a theoretical framework learning 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 time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function 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 unconstrained DTW on 39 of 85 datasets with 3 ties; however, constrained DTW outperforms DiffTW on 48 of 85 datasets with 5 ties.
Abstract:Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to discrete response variables, like class labels. This task is fundamental in domains, including health monitoring, where temporal structure is critical for 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. Moving beyond pairwise alignment, we propose a theoretical framework learning 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 time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function 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 unconstrained DTW on 39 of 85 datasets with 3 ties; however, constrained DTW outperforms DiffTW on 48 of 85 datasets with 5 ties.
39 pages including appendix and references, 8 figures. v2: Corrected a miscalculation in the DTW nearest neighbor accuracy, identified by an author of the UCR archive, Prof. Eamonn Keogh. Updated results show the algorithm performs comparably to unconstrained DTW, but below constrained DTW. Added new discussion on constrained vs. unconstrained DTW
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2606.23472 [stat.ML]
(or arXiv:2606.23472v2 [stat.ML] for this version)
Full text under the publisher's open licence · 265 words
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