Differentiable Divergences Between Time Series
This addresses a key challenge in time series analysis for applications like classification, though it is incremental as it builds on soft-DTW.
The paper tackles the problem of computing discrepancies between variable-length time series by proposing a new differentiable divergence called soft-DTW divergence, which corrects issues in existing methods like non-differentiability and bias, and demonstrates significant accuracy improvements on 84 classification datasets.
Computing the discrepancy between time series of variable sizes is notoriously challenging. While dynamic time warping (DTW) is popularly used for this purpose, it is not differentiable everywhere and is known to lead to bad local optima when used as a "loss". Soft-DTW addresses these issues, but it is not a positive definite divergence: due to the bias introduced by entropic regularization, it can be negative and it is not minimized when the time series are equal. We propose in this paper a new divergence, dubbed soft-DTW divergence, which aims to correct these issues. We study its properties; in particular, under conditions on the ground cost, we show that it is a valid divergence: it is non-negative and minimized if and only if the two time series are equal. We also propose a new "sharp" variant by further removing entropic bias. We showcase our divergences on time series averaging and demonstrate significant accuracy improvements compared to both DTW and soft-DTW on 84 time series classification datasets.