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Update rules for learning in dynamic time warping spaces are based on optimalwarping paths between parameter and input time series. In general, optimalwarping paths are not unique resulting in adverse effects in theory andpractice. Under the assumption of squared error local costs, we show that notwo warping paths have identical costs almost everywhere in a measure-theoreticsense. Two direct consequences of this result are: (i) optimal warping pathsare unique almost everywhere, and (ii) the set of all pairs of time series withmultiple equal-cost warping paths coincides with the union of exponentiallymany zero sets of quadratic forms. One implication of the proposed results isthat typical distance-based cost functions such as the k-means objective aredifferentiable almost everywhere and can be minimized by subgradient methods.

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