Perturbed Dynamic Time Warping: A Probabilistic Framework and Generalized Variants

Dynamic Time Warping (DTW) is a classical method for measuring similarity between time series, but its non-differentiability hinders integration into end-to-end learning frameworks. To address this, soft-DTW replaces the minimum operator with a smooth soft-min, enabling differentiability and efficient computation. Motivated by soft-DTW, we propose perturbed-DTW, a differentiable framework of DTW obtained by adding random perturbations to warping costs and taking the expected minimum. Under Gumbel noise, perturbed-DTW exactly recovers soft-DTW, providing a natural probabilistic interpretation of soft-DTW. We further generalize this framework by extending the Gumbel noise to the broader family of generalized extreme value (GEV) distributions, leading to a new class of soft-DTW variants. Building on this insight, we introduce nested-soft-DTW (ns-DTW), which integrates GEV perturbations into the dynamic programming formulation of perturbed-DTW. This extension induces alignments with tunable skewness, offering greater flexibility in modeling diverse alignment structures. We validate ns-DTW on barycenter computation, clustering, and classification, demonstrating its effectiveness over existing approaches.