  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.
