Aligning Time Series on Incomparable Spaces

TL;DR

This paper introduces Gromov dynamic time warping (GDTW), a novel method for aligning and comparing time series on different, incomparable spaces by focusing on intra-relational geometry, overcoming limitations of traditional DTW.

Contribution

The paper proposes GDTW, a new distance measure for time series on incomparable spaces, and a differentiable version for applications like averaging, generative modeling, and imitation learning.

Findings

GDTW effectively aligns and compares time series on different spaces.

The differentiable GDTW facilitates optimization in various learning tasks.

GDTW outperforms traditional DTW in settings with incomparable spaces.

Abstract

Dynamic time warping (DTW) is a useful method for aligning, comparing and combining time series, but it requires them to live in comparable spaces. In this work, we consider a setting in which time series live on different spaces without a sensible ground metric, causing DTW to become ill-defined. To alleviate this, we propose Gromov dynamic time warping (GDTW), a distance between time series on potentially incomparable spaces that avoids the comparability requirement by instead considering intra-relational geometry. We demonstrate its effectiveness at aligning, combining and comparing time series living on incomparable spaces. We further propose a smoothed version of GDTW as a differentiable loss and assess its properties in a variety of settings, including barycentric averaging, generative modeling and imitation learning.