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This paper addresses learning end-to-end models for time series data thatinclude a temporal alignment step via dynamic time warping (DTW). Existingapproaches to differentiable DTW either differentiate through a fixed warpingpath or apply a differentiable relaxation to the min operator found in therecursive steps used to solve the DTW problem. We instead propose a DTW layerbased around bi-level optimisation and deep declarative networks, which we nameDecDTW. By formulating DTW as a continuous, inequality constrained optimisationproblem, we can compute gradients for the solution of the optimal alignment(with respect to the underlying time series) using implicit differentiation. Aninteresting byproduct of this formulation is that DecDTW outputs the optimalwarping path between two time series as opposed to a soft approximation,recoverable from Soft-DTW. We show that this property is particularly usefulfor applications where downstream loss functions are defined on the optimalalignment path itself. This naturally occurs, for instance, when learning toimprove the accuracy of predicted alignments against ground truth alignments.We evaluate DecDTW on two such applications, namely the audio-to-scorealignment task in music information retrieval and the visual place recognitiontask in robotics, demonstrating state-of-the-art results in both.

Deep Declarative Dynamic Time Warping for End-to-End Learning of Alignment Paths