A General Optimization Framework for Dynamic Time Warping
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Dynamic time warping (DTW) is a method that inputs two time-domain signals and outputs a function that warps time in order to approximately align the two signals together. Despite its popularity, DTW has a problem that has eluded researchers for decades: so-called "singularities," which are sharp irregularities in the time warp function that cause many time points of one signal to be erroneously mapped onto a single point in the other signal. We solve this long-standing problem by reformulating DTW entirely. Here we pose DTW as an optimization problem with several penalty terms in the objective. The first term concerns the time-warped signals, penalizing misalignments, while two additional terms concern the time warping function itself. A penalty on the cumulative amount of warping limits over-fitting, which is similar to "ridge" or "lasso" regularization, and a penalty on the instantaneous slope limits roughness in the time warp function. Different choices of the three objective terms yield different time warping functions that trade off signal fit or alignment and properties of the warping function. The optimization problem we formulate is a classical optimal control problem, with initial and terminal constraints, and a state dimension of one. We describe an effective general method that minimizes the objective by discretizing the values of the original and warped time, and using standard dynamic programming to compute the (globally) optimal warping function with the discretized values. Iterated refinement of this scheme yields a high accuracy warping function in just a few iterations. We include a single, efficient algorithm with linear time complexity and an average runtime that's 50x faster than the state-of-the-art methods on typical problem sizes. Our method and example data are available in an open-source Python package called GDTW.
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