Time Series Classification through Diffeomorphic Time Warping (DiffTW)

Traditional time series classification often relies on Dynamic Time Warping (DTW), which computes similarity through discrete point-to-point alignment. This approach, while effective, ignores the underlying dynamical processes that generate the sequences. The authors propose Diffeomorphic Time Warping (DiffTW), a framework that treats time series alignment as a continuous physical flow, aiming to learn the underlying dynamics that transform one signal into another.

DiffTW models the transformation between two time series as a flow governed by a first-order advection partial differential equation (PDE). By using the method of characteristics, the authors reduce this PDE to a system of ordinary differential equations (ODEs). The velocity field driving this flow is learned within a reproducing kernel Hilbert space (RKHS), parameterized by random Fourier features. To optimize this, the authors frame the problem as an optimal control task, using continuous backpropagation to compute gradients and ensure the alignment satisfies boundary conditions.

DiffTW provides a theoretically grounded dissimilarity measure that avoids the computational limitations of discrete pairwise alignment. In empirical evaluations using electrocardiogram (ECG) data and subsets of the UCR time series archive, DiffTW demonstrated superior classification accuracy compared to standard DTW, outperforming it on 60 out of 86 datasets. The method also exhibits linear scaling with respect to sequence length, suggesting it is well-suited for high-resolution time series data.

By shifting from discrete alignment to continuous dynamical modeling, DiffTW offers a more physically interpretable and computationally efficient way to compare time series. This approach is particularly relevant for domains like health monitoring, where the temporal structure of signals—such as cardiac cycles—is critical for accurate classification and diagnostic performance.