  Dynamic Time Warping (DTW) is used for matching pairs of sequences and
celebrated in applications such as forecasting the evolution of time series,
clustering time series or even matching sequence pairs in few-shot action
recognition. The transportation plan of DTW contains a set of paths; each path
matches frames between two sequences under a varying degree of time warping, to
account for varying temporal intra-class dynamics of actions. However, as DTW
is the smallest distance among all paths, it may be affected by the feature
uncertainty which varies across time steps/frames. Thus, in this paper, we
propose to model the so-called aleatoric uncertainty of a differentiable (soft)
version of DTW. To this end, we model the heteroscedastic aleatoric uncertainty
of each path by the product of likelihoods from Normal distributions, each
capturing variance of pair of frames. (The path distance is the sum of base
distances between features of pairs of frames of the path.) The Maximum
Likelihood Estimation (MLE) applied to a path yields two terms: (i) a sum of
Euclidean distances weighted by the variance inverse, and (ii) a sum of
log-variance regularization terms. Thus, our uncertainty-DTW is the smallest
weighted path distance among all paths, and the regularization term (penalty
for the high uncertainty) is the aggregate of log-variances along the path. The
distance and the regularization term can be used in various objectives. We
showcase forecasting the evolution of time series, estimating the Fr\'echet
mean of time series, and supervised/unsupervised few-shot action recognition of
the articulated human 3D body joints.
