  We propose in this paper a differentiable learning loss between time series,
building upon the celebrated dynamic time warping (DTW) discrepancy. Unlike the
Euclidean distance, DTW can compare time series of variable size and is robust
to shifts or dilatations across the time dimension. To compute DTW, one
typically solves a minimal-cost alignment problem between two 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 gradient can be computed with quadratic time/space complexity
(DTW has quadratic time but linear space complexity). We show that this
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. 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.
