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.

在这篇论文中，我们提出了一种基于动态时间规整（DTW）距离度量的可微学习损失函数，并利用软 DTW 来计算 DTW，该方法可用于平均时间序列和聚类时间序列，且在此任务中比现有基线表现更好。接着，我们通过最小化机器输出时间序列与基准标签的软 DTW 距离来调节其参数。