Differentiable Divergences Between Time Series

TL;DR

This paper introduces a new differentiable divergence for time series, called soft-DTW divergence, which is non-negative, minimized when series are equal, and improves classification accuracy over existing methods.

Contribution

The paper proposes the soft-DTW divergence and a sharp variant, addressing limitations of DTW and soft-DTW by ensuring positive definiteness and better optimization properties.

Findings

Soft-DTW divergence is a valid divergence, non-negative and minimized when series are equal.

The sharp variant further reduces entropic bias, improving performance.

Significant accuracy improvements demonstrated on 84 classification datasets.

Abstract

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.