  Update rules for learning in dynamic time warping spaces are based on optimal
warping paths between parameter and input time series. In general, optimal
warping paths are not unique resulting in adverse effects in theory and
practice. Under the assumption of squared error local costs, we show that no
two warping paths have identical costs almost everywhere in a measure-theoretic
sense. Two direct consequences of this result are: (i) optimal warping paths
are unique almost everywhere, and (ii) the set of all pairs of time series with
multiple equal-cost warping paths coincides with the union of exponentially
many zero sets of quadratic forms. One implication of the proposed results is
that typical distance-based cost functions such as the k-means objective are
differentiable almost everywhere and can be minimized by subgradient methods.
