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Soft-DTW: a Differentiable Loss Function for Time-Series

Author

Listed:
  • Marco Cuturi

    (CREST; ENSAE; Université Paris-Saclay)

  • Mathieu Blondel

    (NTT Communication Science Laboratories)

Abstract

We propose in this paper a differentiable learning loss between time series. Our proposal builds upon the celebrated Dynamic Time Warping (DTW) discrepancy. Unlike the Euclidean distance, DTW is able to compare asynchronous time series of varying size and is robust to elastic transformations in time. To be robust to such invariances, DTW computes a minimal cost alignment between time series using dynamic programming. Our work takes advantage of a smoothed formulation of DTW, called soft-DTW, that computes the soft-minimum of all alignment costs. We show in this paper that soft-DTW is a differentiable loss function, and that both its value and its gradient can be computed with quadratic time/space complexity (DTW has quadratic time and linear space complexity). We show that our regularization is particularly well suited to average and cluster time series under the DTW geometry, a task for which our proposal significantly outperforms existing baselines (Petitjean et al., 2011). Next, we propose to tune the parameters of a machine that outputs time series by minimizing its fit with ground-truth labels in a soft-DTW sense.

Suggested Citation

  • Marco Cuturi & Mathieu Blondel, 2017. "Soft-DTW: a Differentiable Loss Function for Time-Series," Working Papers 2017-81, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-81
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    Cited by:

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    2. Hao Luo & Kexin Sun & Junlu Wang & Chengfeng Liu & Linlin Ding & Baoyan Song, 2019. "Multistage identification method for real-time abnormal events of streaming data," International Journal of Distributed Sensor Networks, , vol. 15(12), pages 15501477198, December.
    3. Ocampo, Lanndon & Abellana, Dharyll Prince & Selle, Julius & Namoco, Rhoda & Maturan, Fatima & Padro, John Kevin & Evangelista, Samantha Shane, 2025. "Dynamical system modelling for picture fuzzy multi-criteria public opinion on transportation infrastructure," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 455-474.
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    5. Mario Flor & Sergio Herraiz & Ivan Contreras, 2021. "Definition of Residential Power Load Profiles Clusters Using Machine Learning and Spatial Analysis," Energies, MDPI, vol. 14(20), pages 1-15, October.
    6. Zhong, Liangjian & Gan, Zuoxian & Yu, Qing & Li, Linchao, 2025. "Micro-mobility behavior resilience analysis in extreme weather events based on a knowledge-informed machine learning approach," Reliability Engineering and System Safety, Elsevier, vol. 264(PB).
    7. Planakis, Nikolaos & Papalambrou, George & Kyrtatos, Nikolaos, 2022. "Ship energy management system development and experimental evaluation utilizing marine loading cycles based on machine learning techniques," Applied Energy, Elsevier, vol. 307(C).
    8. Brijnesh Jain & Vincent Froese & David Schultz, 2023. "An average-compress algorithm for the sample mean problem under dynamic time warping," Journal of Global Optimization, Springer, vol. 86(4), pages 885-903, August.
    9. Westermann, Paul & Deb, Chirag & Schlueter, Arno & Evins, Ralph, 2020. "Unsupervised learning of energy signatures to identify the heating system and building type using smart meter data," Applied Energy, Elsevier, vol. 264(C).

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