Optimal Warping Paths are unique for almost every pair of Time Series

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.