Aligning Time Series on Incomparable Spaces
This addresses a limitation in time series analysis for domains with incomparable data spaces, though it appears incremental as it extends DTW with a known geometric approach.
The paper tackles the problem of aligning time series that exist in incomparable spaces where traditional dynamic time warping (DTW) fails, by proposing Gromov dynamic time warping (GDTW) and a smoothed version for differentiable applications, demonstrating effectiveness in tasks like barycentric averaging and generative modeling.
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.