Fast Gradient-Based Approximation of Continuous Dynamic Time Warping
Abstract
Dynamic Time Warping (DTW) is a common choice for measuring the discrepancy between time series. It proceeds by finding an optimal warping between timestamps through dynamic programming, which can be slow and is not differentiable. Consequently, several variants have been proposed. Among them, Soft-DTW proposes a smooth formulation of DTW, amenable to gradient computation, while Continuous-DTW (CDTW) introduces a continuous formulation through parametrizable warping functions. Yet, computational costs remain high for both methods and impractical for large datasets in modern machine learning pipelines. This paper revisits CDTW in several ways. After establishing its differentiability under mild conditions, we introduce FastCDTW, which is an efficient approximate solver with two different computational strategies: an exact closed-form scheme for piecewise-linear warping functions, and an unbiased Monte Carlo estimator for smooth ones, such as implicit neural networks. Both strategies provide gradients with respect to the warping function and the input time series, and the resulting algorithm is GPU-friendly. We demonstrate its benefits through its integration into more complex learning problems, including time series averaging, classification and deep domain adaptation.
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BibTeX
@misc{houedry2026fast,
title = {{Fast Gradient-Based Approximation of Continuous Dynamic Time Warping}},
author = {Pierre Houedry and Romain Tavenard and Nicolas Courty and Romaric Gaudel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13129},
eprint = {2610.13129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13129},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Fast Gradient-Based Approximation of Continuous Dynamic Time Warping
AU - Pierre Houedry
AU - Romain Tavenard
AU - Nicolas Courty
AU - Romaric Gaudel
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13129
UR - https://arxiv.org/abs/2610.13129
AB - Dynamic Time Warping (DTW) is a common choice for measuring the discrepancy between time series. It proceeds by finding an optimal warping between timestamps through dynamic programming, which can be slow and is not differentiable. Consequently, several variants have been proposed. Among them, Soft-DTW proposes a smooth formulation of DTW, amenable to gradient computation, while Continuous-DTW (CDTW) introduces a continuous formulation through parametrizable warping functions. Yet, computational costs remain high for both methods and impractical for large datasets in modern machine learning pipelines. This paper revisits CDTW in several ways. After establishing its differentiability under mild conditions, we introduce FastCDTW, which is an efficient approximate solver with two different computational strategies: an exact closed-form scheme for piecewise-linear warping functions, and an unbiased Monte Carlo estimator for smooth ones, such as implicit neural networks. Both strategies provide gradients with respect to the warping function and the input time series, and the resulting algorithm is GPU-friendly. We demonstrate its benefits through its integration into more complex learning problems, including time series averaging, classification and deep domain adaptation.
ER -