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