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

TL;DR

This paper proves that for most pairs of time series, the optimal warping path in dynamic time warping is unique, ensuring differentiability of related cost functions and improving learning algorithms.

Contribution

It establishes that optimal warping paths are almost surely unique under squared error costs, resolving issues caused by path non-uniqueness in DTW-based learning.

Findings

Optimal warping paths are unique almost everywhere.

Distance functions like k-means are differentiable almost everywhere.

Sets of non-unique paths are measure-zero, negligible in practice.

Abstract

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.